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OHJ: The Hallquist Review

Cover of Richard Carrier's book On the Historicity of Jesus. Medieval icon image of Jesus holding a codex, on a plain brown background, title above in white text, author below in white text.This week I am doing a series on early reviews of my book On the Historicity of Jesus. If you know of reviews I don’t cover by the end of the first week of July, post them in comments (though please also remark on your own estimation of their merits).

-:-

One of those early reviews posted is by Chris Hallquist (at The Uncredible Hallq for Patheos), a notable atheist author who has a master’s in philosophy from Notre Dame. His review is billed as only “initial thoughts” and therefore might be revised or expanded in future posts. If so, I’ll blog those and add links at the bottom here (please let me know if he blogs again on the subject so I can do that). For now, here is my commentary on what he has posted so far.

Hallquist has overall good impressions of the book. But he goes on to make statements that suggest a poor understanding of how probability works (ironically, since he makes the point himself that people often have a poor understanding of how probability works).

Here is what I mean:

  • “But if most versions of mythicism can be shown to be very improbable, shouldn’t that lower the overall probability of mythicism?”

That’s like saying if most theories of historicity are improbable, shouldn’t that lower the overall probability of historicity? Yet most theories of historicity are improbable. In fact, that they are is a mathematically necessary truth. For example, if there are ten competing theories of historicity (and that’s not far from the case: Proving History, ch. 1), and all are equally likely, and historicity overall is virtually 100% certain (and it doesn’t matter whether at this stage we are talking about the prior of the posterior probability), then every theory of historicity is improbable–because then every single one has about a 10% chance of being true, which is indeed improbable. Yet historicity overall remains 100% certain.

On the other hand if one theory of historicity is far more plausible than the others (and I would say this is true of the Ehrman model), let’s say it is 90% likely and all others fill the remaining 10% or so, then “most versions of historicity can be shown to be very improbable,” because now the remaining nine theories occupy that 10%, so if they are all equally likely, then most theories have around a 1% chance of being true, which is indeed very improbable. Yet historicity would still be virtually 100% likely (and Ehrman’s theory would still be 90% likely). Thus, Hallquist does not appear to understand how probability works. We’ll see more examples of that shortly.

Imagine saying, “If most theories of motion have been improbable, shouldn’t that have lowered the overall probability of Newtonian mechanics?” As just explained, that is a failure to understand how a prior probability is constructed. The reference classes have to be distinguished: some false theories of motion (e.g. Epicurean; Stratonian) were in fact more probable than others (e.g. Empedoclean; Aristotelian). What you want to compare are not crazy theories with plausible theories, but crazy theories with crazy theories and plausible theories with plausible theories. The existence of crazy theories does not reduce the probability of plausible theories. That’s what calling them plausible means: they are no longer in the reference class of improbable (“crazy”) theories.

For example, in the discussion Hallquist is referring to (OHJ, pp. 52-55) I rule out the hypothesis that Jesus didn’t exist yet the first apostles started right out of the gate claiming everything in the Gospels really happened in Jerusalem and Galilee. I hardly need explain why that is massively improbable. But it’s being massively improbable has no relevance at all to whether the hypothesis I do defend is–that Jesus didn’t exist yet the first apostles claimed to be receiving visions of a celestial Jesus. If Jesus didn’t exist, then the latter is highly probable, for precisely the same reason the former is highly improbable: I accumulate considerable background evidence in favor of the second theory’s prior probability (in chs. 4 and 5); nothing exists comparably supporting even a noticeable prior for the “total instant lie” hypothesis (I can’t think of a single analogous example in all of history; not even the infamous liar Joseph Smith attempted that).

That no background evidence supports the “total instant lie” hypothesis does not allow the inference that no background evidence supports the “visions of a celestial being” hypothesis.

Instead, what we have to do (and logic requires this, as I explain in Proving History, pp. 166-67, 245-55) is divide the “probability space” among all theories within a covering theory (all theories that can be true if Jesus did not exist; literally: all logically possible theories, no matter how absurd). How much of that space is occupied by each theory? Some are so absurd they occupy very little of it (like the “total instant lie” hypothesis). Some are so plausible they occupy a lot of it (like the “visions of a celestial being” hypothesis), representing the fact that the latter is many times more initially probable than the former. I explain this in detail in OHJ, pp. 27-29, and I must assume Hallquist didn’t read those pages, or didn’t understand them.

For more on how prior probabilities work see my commentary on Lataster’s review.

  • “In Bayesianism, something is counted as ‘evidence’ for a hypothesis if it raises the probability of the hypothesis.”

Hallquist claims I violate this principle. He gives no examples. I am unaware of any.

Ironically, he thereby violated this principle.

(It’s unclear to me that Hallquist actually understands what it means for e to raise the probability of h; if e is just as likely on h as on ~h, even if e is 100% expected on h, then e does not raise the probability of h; it is therefore not evidence for h; so far as I know–and I was very careful–that is all I ever argue anywhere in OHJ.)

  • “Why not, for example, put Jesus in the reference class of apocalyptic preachers, faith healers, and exorcists?”

This statement suggests to me that Hallquist did not read my book. Because I actually address this directly and in detail, with mathematical demonstrations. I even call it “The Alternative Class Objection,” and it is even listed in the table of contents as such! See OHJ, pp. 245-46. I demonstrate there that even starting with other reference classes gets you the same result–because you always have to put the Rank-Raglan data back in. I explain the reason for this in PH, pp. 240-42. I give a direct demonstration of it in OHJ.

The worst kind of criticism is one that is already devastatingly refuted in elaborate detail in the very book being reviewed. And the critic doesn’t even know it.

But I must say the following criticism is even worse, because it is a seriously embarrassing thing for anyone to say:

  • “[A]rguing that the prior probability of a historical Jesus is low because Jesus’ story shares many features in common with that of mythic heroes strikes me as extremely dubious. Consider, who is the following paragraph describing?”

Hallquist then gives an example of a historical Rank-Raglan hero. It does not appear he actually does, since I only count those who score above ten out of the twenty two criteria, and Hallquist’s example does not appear to do so. But let’s set aside that gaffe, because it’s just an example of not paying attention to what I actually (and carefully and in detail) argue. The far more galling mistake here is that Hallquist just basically said the same thing as this:

  • “Arguing that the prior probability of dying from a vaccine shot is low because lots of people get those shots without dying strikes me as extremely dubious. Consider, I know a guy who died from a vaccine shot.”

Do I really need to explain what’s wrong with that statement?

Only someone who didn’t understand what a probability was could say such a ridiculous thing. Obviously when I say the probability is low, I am actually agreeing that some historical people meet the condition–if they didn’t, I wouldn’t say the probability is “low,” I’d say it was zero. (Rosson made the same mistake, but much less obviously and thus much less embarrassingly.)

And in fact in my math (in chapter 6) I allow for between 1 and 4 people who score above ten on the Rank-Raglan scale to have been historical. So Hallquist is actually stating as a criticism something I actually stated already in the very book he claims to be responding to–only I didn’t make a boner mathematical mistake out of it, like he did. This should not instill much confidence in his ability to reliably critique the rest of my book.

The most amusing irony is that I even did almost exactly the same thing he did: only my example wasn’t Kim Jong-Il, it was Haile Selassie (OHJ, pp. 18-20). Who is actually a better example.

I should also point out that, except when dealing with cross-cultural universals like human biology or economics or social dynamics, we can no longer use modern examples as relevant to the reference class for Jesus (because without a sound reason, we can’t use modern frequencies of anything distinctly cultural as the ancient frequency of it: I explain this mistake in PH, pp. 18, 245). But even if we played that game, Hallquist would lose. Because Superman also fits the Rank-Raglan profile, as do Anakin Skywalker, Optimus Prime, Aragorn, and Captain Kirk: all score above 10. But it would be as invalid of me to use those heroes to argue Jesus is less likely historical, as it would be of Hallquist to use Kim Jong-Il to argue the reverse. The context of all these examples are no longer sufficiently applicable. Although note what would happen if we did what Hallquist wants, and counted all modern examples scoring above 10. Do the math. If you know what a “probability” is, you’ll laugh.

-:-

For a complete list of my responses to critiques of OHJ, see the last section of my List of Responses to Defenders of the Historicity of Jesus.

Comments

  1. Giuseppe says

    The book is arrived (finally!) today, but when I read Hallquist ”review”, already from reading these:

    Carrier’s framing of the historicist hypothesis is extremely minimalistic, but gives a more detailed version of the mythicist hypothesis. …. I fear Carrier may have stacked the deck in his favor, because we know that psychologically, adding details to a story often makes it feel more plausible, even when logically this makes no sense.

    I became suspicious with the author.

    Because the conjunction fallacy is a typical error of historicists, usually: to suppose the existence of a personality worthy of a ”historical Jesus” is already an extremely complex hypothesis.

    I wonder why it’s absent this observation in your post (it’s another case where the accuser is guilty of the accusation).

    • says

      He’s reacting to the fact that I list three features defining minimal historicity (OHJ, p. 34), and five defining minimal mythicism (OHJ, p. 53). He confuses this as meaning the latter is more ad hoc than the former, even though I explain immediately in the book why it’s not.

      But you are correct to note as well that “historicity” covertly relies on countless ad hoc excuses for why the evidence doesn’t fit. I discussed this in my response to Bermejo-Rubio, since he explicitly assumes the contrary as his argument for historicity, so the fact that even he was overlooking all the assumptions being snuck in was a point worth making there. But in the book, I also address aspects of this, e.g. in chs. 8 and 11, where I talk about some of the hidden assumptions historicists pile on to the basic assumption of historicity, in order to explain away problematic evidence.

  2. millssg99 says

    Why is everyone so worried about prior probability? Regardless of low prior probability wouldn’t actual evidence of historicity overcome that in a real historical person? Is the reason everyone obsesses about prior probability because their evidence of historicity is so embarrassingly weak?

    • says

      It’s more that there is an anti-Bayesian apologetic in the culture-sphere that people are vaguely aware of such that they think “subjective priors” makes all Bayesian arguments worthless and thus they attack there the moment the word “Bayes” is uttered, not really comprehending what they are doing or even what a prior probability actually is.

      But yes, indeed, even a low prior can be overwhelmed by evidence (I give an example in OHJ, pp. 252-53: even if Julius Caesar maxed out the Rank-Raglan scale, the evidence for him is vast, and would hugely overwhelm even a 6% prior). So they shouldn’t be afraid of low priors. They should be afraid of low likelihood ratios. But that requires actually understanding Bayesian logic.

      Your insight is correct: the evidence for Jesus is so poor that a low prior becomes terrifying; whereas if we had good evidence for Jesus, a low prior wouldn’t worry anyone.

  3. busterggi says

    “Why not, for example, put Jesus in the reference class of apocalyptic preachers, faith healers, and exorcists?”

    For the same reason we can’t put him in as a Dwarf, Elf or Halfling?

    Sorry, knee-jerk reaction.

    • says

      Yes, that is a bit uncharitable.

      Jesus could fall into those classes, when properly framed in a non-circular way. I discuss an example in Proving History, pp. 240-41: legendary Rabbis (some of whom are historical, some of whom are mythical). We just don’t have enough data to use that class (we don’t know how many legendary Rabbis really existed or didn’t). But Jesus would also belong to many other comparable classes (legendary humans, legendary Jews, legendary inhabitants of the Roman Empire; and legendary wizards, demigods, savior deities; and thus also legendary apocalyptic preachers, faith healers, and exorcists). We just, again, don’t have any usable data for them (we don’t know how many really existed or didn’t, or even how many there were). That every claim occupies countless overlapping reference classes, and how to pick one in light of that, is something I devote a whole section to in PH, pp. 229-56.

  4. says

    This is somewhat unrelated but there’s a debate over the historical possibility of the size of Jesus’s family if it were biological (which involved James McGrath among all people).

    Original post – the anonymous blogger finds a passage in Matthew which suggests that Jesus’s “family” consisted of 9+ and using various studies argues that if it were meant as a biological family it would be unlikely (rarity of such a family) to impossible (because of proposed family size limits).

    Reply by McGrath – Thinks the ancient estimates of population size are too old to be used (and apparently conflated the two arguments in the original posts via the average).

    Reply by blogger – Points out the conflation, argues that the ancient economy did not change making it relevant to use old estimates of population limits.

    Update by blogger – updates his population probabilities.

    Seeing as how you’re background is in ancient history would using population estimates in the way the blogger did be valid?

    • says

      That’s all fascinating. Thanks for including all those links and summaries. Very helpful.

      The problem I have with that is not the math or demography (which looks correct; McGrath is wrong, or at least moot) but with the assumption that an unusually large family is the converse of not-existing. Obviously hundreds of thousands of families were that large (by the blogger’s own math). And there are other alternative hypotheses (Jesus existed but his family’s size was exaggerated; Jesus’s siblings are by multiple mothers/fathers). So just because person x had an unusually large family, that does not increase the likelihood that x didn’t exist (at least not significantly). You’d have to show that fake persons have unusually large families at a higher rate than real persons do, and the difference in those two rates is the likelihood ratio that would generate your prior, not the rate of a historical person having a large family.

      Which would require determining both rates. The blogger did that for real persons; but how would we do it for fake persons? Not enough data, and IMO what data we have could argue the reverse (fake persons tend to have small or no families, and thus have unusually large families at a lower rate than historical persons).

      In the end, either way, the Rank Raglan data still go in, and thus still get us to the same prior–unless we could show that high scoring members of that class have statistically different family sizes whether historical or not, and (a) we cannot determine any to be historical (so we have no data) and (b) even if we could, there are only 15 members, which is far too small. With so few members, any variations between historical and non-historical members vis-a-vis family size could be entirely explained by random chance–unless the deviation was very consistently extreme. Therefore, such a differential would be of no use to us. The margins of error would dissolve any likelihood ratio you attempted to produce.

  5. says

    “I demonstrate there that even starting with other reference classes gets you the same result–because you always have to put the Rank-Raglan data back in.”

    Could you explain why Rank-Raglan data would always be needed? If someone were to use the dataset Hallquist suggests, some preachers and mystics would score very high (like Jesus and Moses) but most would probably not score high if at all (outside of maybe the occasional 1-2 for a mysterious death and/or sepulchre). It seems clear that such an uncorrelated sample would not be relevant for rank-raglan data.

    And I don’t think this is the same as demonstrating which rank-raglan score would be a cut-off because that presupposes the utility of the score, in this case we would be including figures regardless of the score.

    That aside, I think there are good reasons for using the score just not that it’s inherently needed no matter what measure you use.

    • says

      Could you explain why Rank-Raglan data would always be needed?

      Bayes’ Theorem follows the Rule of Total Evidence. That means all data goes in (b + e = all human knowledge whatever).

      So it doesn’t matter where you start. It all gets counted.

      I explain the effect of this in the section on the Alternative Class Objection, as well as in the main discussion in ch. 6 (so, § 3 and 5). I even give an example (Josephan Messiahs as an alternative class) and show it makes no difference.

      What is convenient about the RR scale is that it allows us to identify a large set of members (at least 15) who are peculiarly non-conforming to the general population (unusually high rate of non-existence). If Jesus did not belong to any such class, the prior probability of his existing would be higher (although note my discussion of the Jesus-as-name class in ch. 6 § 3; its only defect is poverty of usable data).

  6. says

    To repeat what I wrote on Chris Hallquist’s blog: The whole exercise of applying the hero class appears very dubious to me. If it were defined by perhaps three to five very clear criteria, all of which need to be fulfilled, then we would be talking. But having a mere 22 sometimes very fuzzy criteria of which less than half (!) need to be fulfilled – surely that leaves too much room for motivated reasoning?

    From a plain reading of the criteria, several of the participants in the Wars of the Roses fulfill around two thirds of them, because opponents would try to murder them when they were young, they would be raised in exile, they would return at the head of an army, they were descended from kings, they would marry into related European royalty, defeat a strong enemy, succeed their fathers, rule for a time uneventfully, make laws, be deposed, still have a grave in an important church, and so on. The same goes for medieval royalty in France and Germany, and all of them were historical.

    In addition, what does it actually show if somebody checks those criteria even if they are interpreted more strictly? Merely that followers may have mystified an utterly mundane historical figure after their death.

    Finally, I find it hard to take serious any attempt to turn these considerations into numerical probabilities. Sometimes it is okay to admit that we cannot come up with something more precise than a hunch.

    • says

      surely that leaves too much room for motivated reasoning?

      I address this in the book. If it were just motivated reasoning, then there should be many more historical persons in the class. Yet somehow there aren’t (indeed, there do not appear to be any). That can’t be a coincidence. Therefore, the class is real. It is not a mirage. And that remains a fact, regardless of why it is a fact (see ch. 6, § 4).

      By contrast, if the class contained only two members, then you could fabricate a mirage class. But for there to be so many members, who score so high, is beyond statistical probability as a consequence of chance. Leaving the question of whether a member’s matching it reflects the larger population, or peculiarly doesn’t. That it peculiarly doesn’t is what makes the class significant.

      As to hunches, I discuss them in ch. 6, and show why the RR class is in fact better than a hunch. As long as we introduce reasonable margins of error. Which I do.

    • says

      I am sorry but I do not understand the answer because a major point of my comment (as well as of Chris Hallquist’s Kim Jong-Il example) is that myriads of historical people easily fulfill ten to twelve criteria and often more. I find it hard to believe that you simply deny this because various combinations of criteria 2, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21 and 22 are standard procedure for feudal lords under unstable political conditions, 8 is true for virtually every person who has ever lived on this planet, and the only supernatural criteria (1, 3 and 4) can simply be tacked onto a historical person by their followers.

      Really the class is teeming with historical people unless we start quibbling about application of the criteria, which is where the motivated reasoning comes in. Conversely, it does not seem appropriate to count 20th century fictional superheroes into the class because that comparison presupposes that the gospels fall into a 1st century tradition of fantasy novels, thus begging the question.

      Utility of the hero class category or not, it all boils down to this: how would you differentiate a myth and a mythologized mundane person who actually existed? If the followers of some third rate failed doomsday preacher of today adopted the preexisting RR archetype and used to make him sound more holy, would historians in two thousand years and after the next dark age be able to tell the difference?

    • says

      myriads of historical people

      Like who? (I assume this is hyperbole; myriad means 10,000)

      This would be a valid point, if there were any such people in antiquity, then that would change the ratio. So who do you have in mind?

      BTW, there is no such thing as a “feudal lord” in antiquity; only culturally relevant contexts matter. But note that if it were true that scoring above 10 was “standard procedure for feudal lords under unstable political conditions” then why are there no examples of historical persons meeting them in antiquity? You can’t say we should expect this to commonly be the case, when we observe that in fact it is never the case. Reality is trumping your expectation here. And we have to follow reality, not your armchair assumptions.

      …how would you differentiate a myth and a mythologized mundane person who actually existed?

      The evidence. That’s chs. 7-11.

      If you don’t have evidence, then the answer is: you can’t.

      I specifically discuss this point with the Haille Selassie example in chapter 2. The converse example is Roswell (index).

    • millssg99 says

      I have a challenge for your 3rd rate preacher scenario. Go find the 3rd rate preacher whose followers and promoters never quote his preaching or describe events in his life.

    • says

      Yes, that would fall under “the evidence” category. Hence it’s worth reminding people that any low prior can be overcome by evidence. So it’s the lack of evidence, indeed even the presence of contrary evidence, that is precisely the problem. Bitching about the fact that the guy looks mythical, therefore he is initially more likely to be, is just a handwaving tactic to avoid facing the problem of evidence. I make exactly this point about Alexander the Great in OHJ: even if his prior were 6%, his historicity would remain certain. So a low prior isn’t the problem.

  7. Bernard says

    Dr. Carrier,

    “yet the first apostles started right out of the gate claiming everything in the Gospels really happened in Jerusalem and Galilee.”

    From where did you get that? It is a straw man in my view, which certainly makes your mythicism highly probable, and historicity out of question, except if Jesus was very much according to the Gospels. And we know it was not the case.
    Are you addressing fundies’ beliefs? Certainly it cannot be minimal historicity.

    “I demonstrate there that even starting with other reference classes gets you the same result–because you always have to put the Rank-Raglan data back in.”

    You use a gospel (gMatthew) written two generations after Jesus’ crucifixion. Sure, we are going to see a lot of mythical accretions. But gMark, written earlier, would score fairly low on your Rank-Raglan list.

    BTW, I did order your book.

    Cordially, Bernard

    • says

      Mark alone gives Jesus a score of 14. So I don’t require Matthew. Although by mainstream opinion they differ in date by as little as ten years.

      Again, if you would actually read the book you are talking about, you would know this. You would also know I discuss the accretion model and the issue of pace of legendary development. And the fact that sometimes this happens to historical persons. And so on.

      You are just acting like the crank that you are and ignoring all my actual arguments and discussion.

      Indeed, you are so clueless you don’t realize that I actually said “And we know it was not the case” in re: what you call a straw man. You seem not to grasp the logic of what we are even discussing here: Hallquist confusingly thought even absurd theories like that should lower the probability of non-absurd theories. My point in giving an example of an absurd theory was to show the error in his thinking. That you actually thought I was proposing such a theory as plausible shows me that you don’t have a clue what we are talking about here.

      As usual.

  8. Phillip Hallam-Baker says

    OK so today the eldest (13) stated that he is not a Christian because he does not believe in Santa Claus…

    After a long explanation that Santa Claus is not a part of actual Christian theology I was forced to admit that there is actually more evidence for him than for the historical Jesus.

    And if you were to examine the evidence say 100 years later after the Koch bros have caused the destruction of human civilization with their climate change denial you would be pretty hard pushed to find statements to the effect that he is mythological.

  9. messing says

    Only someone who didn’t understand what a probability was could say such a ridiculous thing. Obviously when I say the probability is low, I am actually agreeing that some historical people meet the condition–if they didn’t, I wouldn’t say the probability is “low,” I’d say it was zero

    Someone who understands probability theory might recognize that a probability of 0 doesn’t necessarily mean impossible and a probability of 1 doesn’t mean certain. The matter is addressed in particular at times 9:35-10:01 & 10:35-11:18 in the lecture 2. Conditioning and Bayes’ Rule.

    How much of that space is occupied by each theory? Some are so absurd they occupy very little of it (like the “total instant lie” hypothesis).

    Interestingly, as someone with a familiarity of calculus such as yourself knows, for some sets of all possible outcomes (or for some probability functions) every, single possible outcome/event/hypothesis in the probability space takes up 0 space. Such other nuances of probability abound. There is a neat little text on some of these which I mention because it contains the following quote:

    Use of Bayes’ theorem should not be confused with Bayesianism, a controversial viewpoint on the nature of probability associated with subjectivism and the claim that correct statistical inference depends crucially on assessment of prior distributions. The most avid frequentist has no quarrel with Bayes’ Theorem proper.

    p. 8 of Eckhardt, W. (2013). Paradoxes in probability theory (SpringerBriefs in Philosophy). Springer.

    Your discussion of “priors” as well as BT tends to conflate BT and the Bayesianism your sources in Proving History and thus conflate prior probability with prior distributions, ending up with explication of neither but a confusion of both. How much of this is because you had to simplify I do not know, but it is odd that in a book devoted to defending your formal, logical basis for the method used in On the Historicity of Jesus, you fail to make even so basic a distinction between validity and soundness, rendering the entirety of your “formal proof” (beginning on p. 106) meaningless (infinitely many valid arguments can contradict any “logically proven theorem”, because valid arguments need not be true). Your reference to “a complete proof of the formal validity of BT” (i.e., Papoulis’) isn’t simply another problematic use of “validity” but is a proof of a theorem you don’t use. This basic failure to treat formal logic as just that (formal logic, not laypersons’ terms for notions associated with logic) continues in On the Historicity of Jesus, where you present a hypothesis and its negation using formal notation but translate the proposition corresponding to the negation of the hypothesis as something which isn’t logically equivalent to the negation of your hypothesis (p. 30). The negation of “Jesus was a historical person mythicized” isn’t “Jesus was a mythicized person historicized”, once again showing that formal logic (in addition to probability theory) isn’t as clear cut as you seem to think. My favorite teaching example of the ways in which negation in formal logic can be counterintuitive comes from Hubbard & Hubbard’s Linear Algebra, Vector Calculus, and Differential Forms:

    Clearly the opposite of the (false) statement, ‘All rational numbers equal 1,’ is the statement, ‘There exists a rational number that does not equal 1.’ However, by the same rules, the statement, ‘All seven-legged alligators are orange with blue spots’ is true, since if it were false, then there would exist a seven-legged alligator that is not orange with blue spots

    .
    Your references to formal logic followed by informal explanations and informal uses of logical terms as well as the conflation of various interpretations, approaches, and philosophies of probability render difficult the task of differentiating when you are simplifying and should be understood as doing so and when you mean your uses of technical terms to be taken as such.

    • says

      Someone who understands probability theory might recognize that a probability of 0 doesn’t necessarily mean impossible and a probability of 1 doesn’t mean certain. The matter is addressed in particular at times 9:35-10:01 & 10:35-11:18 in the lecture 2. Conditioning and Bayes’ Rule.

      Let’s not confuse the public here. What he is talking about is an erroneous probability assignment. That’s not the same thing. If the probability really were zero, it really would be impossible. If you then observe it, what you learned is that it did not have a zero probability. Likewise on the other end.

      This is why we work always with margins of error (so we never say or assume the probability is x, but between x and y, and therefore we can never say it’s zero, because when x is zero, y is necessarily nonzero). We must always acknowledge that nothing has a zero probability (except things like cognito ergo sum). I discuss this in Proving History, pp. 23-26. That is actually the point being made in the video you point to. Maybe you didn’t understand it?

      Interestingly, as someone with a familiarity of calculus such as yourself knows, for some sets of all possible outcomes (or for some probability functions) every, single possible outcome/event/hypothesis in the probability space takes up 0 space.

      You are talking about infinitesimals. And you are basically wrong.

      Not only are infinitesimals different from zero (there are in fact infinite cardinalities of infinitesimals, e.g. some are larger or smaller than others, and there is a whole system of arithmetic for this, and zero is only one of these numbers: for resources see my comment here), but an infinite partition of a space is not necessarily wholly comprised of infinitesimals, and in this case (dividing prior probability space) almost never is.

      For example, if a theory takes up 90% of the space, the remaining space could be divided among infinitely many theories that each occupy an infinitesimal proportion, but the first theory still occupies 90%, not an infinitesimal. If there is a theory in the remaining space that occupies 1%, it still occupies 1%, which is still not an infinitesimal. Only the 9% space that is left might then be occupied by infinitesimal spaces, and then those are only the theories that are maximally improbable (and thus you never have to consider them, unless truly extraordinary evidence for them exists).

      Both points combine: by the laws of calculus, that 90% space is actually the sum of infinitely many theories (all the variants that still share the same general attributes). Because infinitesimals can sum to finite numbers. If they didn’t, calculus wouldn’t exist as a thing. The fact that infinitesimals sum to nonzero numbers was discovered and used even by Archimedes 2200 years ago.

      Use of Bayes’ theorem should not be confused with Bayesianism, a controversial viewpoint on the nature of probability associated with subjectivism and the claim that correct statistical inference depends crucially on assessment of prior distributions. The most avid frequentist has no quarrel with Bayes’ Theorem proper.

      Note that this debate is more smoke and mirrors than anything substantial. This I demonstrate in the last sections of ch. 6 of Proving History.

      …you fail to make even so basic a distinction between validity and soundness…

      n.b. “arguments must be logically valid, and factually sound” Proving History, p. 45.

      …rendering the entirety of your “formal proof” (beginning on p. 106) meaningless (infinitely many valid arguments can contradict any “logically proven theorem”, because valid arguments need not be true).

      Which is relevant to that proof how?

      …where you present a hypothesis and its negation using formal notation but translate the proposition corresponding to the negation of the hypothesis as something which isn’t logically equivalent to the negation of your hypothesis (p. 30).

      Incorrect. I explain in detail in OHJ that the negation includes other hypotheses, and then argue their priors are smaller than a tenth of a percent and thus fall below the resolution of the mathematics to be used in the book (pp. 53-55). I therefore properly account for them, and properly remove them. You can certainly re-include them in the math, and thus produce a needlessly complex and time-wasting way to get the exact same result. But why would you want to do that?

      The negation of “Jesus was a historical person mythicized” isn’t “Jesus was a mythicized person historicized”

      Funny how that is exactly what I say as well: OHJ, p. 30.

      Don’t pretend you are schooling me, when the very page you are citing actually says the very thing you claim to be schooling me on.

      It sounds to me either that you are being deliberately disingenuous (why?) or you have no idea what the word “efficient” means. Do you really want vast pages of needless complexity added to an argument, when the end result will exhibit no relevant difference? (Why?)

      More to the point, even though my book specifically explains what I am doing (that I am making the approach more efficient, by carefully and properly ruling out the need to account for other hypotheses, logically and mathematically, and doing so in an entirely valid manner), why do you act like it doesn’t?

    • Anonymous Coward says

      I agree with everything in your reply to Messing except where you say this:

      “If the probability really were zero, it really would be impossible.”

      It’s my understanding that (on the standard interpretation anyway?) a probability of zero means “either it never happens or it almost never happens,” where “almost never” is a technical term: http://en.wikipedia.org/wiki/Almost_surely

      From the wikipedia article just cited: “In probability theory, one says that an event happens almost surely (sometimes abbreviated as a.s.) if it happens with probability one.”

      And the converse of “almost surely” is “almost never.”

      Isn’t that in contradiction with what you said (that I quoted above)?

    • says

      Yes, but again, you are referring the use of the word zero as a stand-in for infinitesimal (and 1 as a stand-in for infinitesimally close to 1). That is not what zero or one means in logic, finite arithmetic (vs. transfinite arithmetic, which follows different rules), or in common human parlance. Hence using it that way can only cause confusion. I recommend not doing so. Stick with what you mean: infinitesimal or (and this would be more accurate, since you never really know that a probability is infinitesimal), “vanishingly small.” Likewise, say “almost surely” rather than “100% certain.” The former meaning is clear; the latter is not, as it can easily be mistaken for saying “logically necessary” / “no possibility of being wrong.” Even by mathematicians, despite their having peculiar dialectical uses of these words unfamiliar to the public.

    • dthunt says

      Richard Carrier said:

      I explain in detail in OHJ that the negation includes other hypotheses, and then argue their priors are smaller than a tenth of a percent and thus fall below the resolution of the mathematics to be used in the book (pp. 53-55). I therefore properly account for them, and properly remove them. You can certainly re-include them in the math, and thus produce a needlessly complex and time-wasting way to get the exact same result. But why would you want to do that?”

      The ~h ~= myth approximation irritated me as well (gut reaction, cannot be helped), though I follow your argument.

      I feel like the proper response to taking an issue with this approximation for your myth prior is either showing convincingly that the RR criteria support historicity more than you argue, or by arguing that you’ve misjudged the amount of the remaining hypothesis space that you’ve discarded by roughly equating ~h to myth; probably by showing several plausible examples in the discard space and providing some reason to believe that they are relatively on the same order as myth, given background.

      I have read Jaynes’s book, and found his review of ‘paradoxes’ to be enlightening. Is Eckhardt’s Paradoxes worthwhile?

    • says

      Note that if someone showed one of the other myth hypotheses to have a higher prior, that would not rescue historicity. The prior for historicity remains unchanged.

      You do realize that, yes?

      (It seems several commenters here are unaware of that.)

      So, for example, if we concluded that the Doherty thesis and (let’s say) the Atwill thesis (!) were initially equally likely, that would mean on the a fortiori estimations, the priors break down as ~33% historicity, ~33% Doherty, ~33% Atwill, which entails ~33% historicity and ~67% myth. Exactly the same result I get in the book.

      Then when we applied the likelihoods, maybe the Atwill thesis would plummet to 1%, but historicity would remain roughly where I found it, while the Doherty thesis would be reduced only by roughly 1%. Or if the Atwill thesis held up, we’d have a posterior of historicity around 32%, Doherty around 34% and Atwill around 34%, so we’d know mythicism was still twice as likely as historicity, but not which version of mythicism was true.

      But the amount of analysis would then have to increase a third, because now you have to analyze the likelihood of all the evidence on the Atwill thesis, as well as the Doherty thesis.

      Again, if you want to undertake all that unnecessary work, you are welcome to, and my book shows you how. But we can tell even before starting that it won’t make a difference.

      So why bother?

      The Atwill thesis does not have a plausible prior, nor does it get a favorable likelihood ratio.

      So why do we need to waste time considering it?

      Ditto every other alternative. Unless, of course, you do have a plausible alternative. In which case I welcome its development in Bayesian terms (and even say so in OHJ), to see how it compares to historicity.

      But that isn’t going to affect the probability of mythicism overall. Because that will always be the sum of the posterior probabilities of the Doherty thesis and every other mythicist thesis.

      P.S. On Eckhardt’s Paradoxes in Probability Theory, I suggest also reading Drescher’s Good and Real. But though Eckhardt’s book is handy on the subject, those paradoxes don’t much relate to the present question of modeling the historicity problem (unless you see a link between one of them and it, in which case let me know your thinking).

    • Latverian Diplomat says

      IMO, the lecturer you linked to is doing a poor job on this point.

      It’s important to note that he’s talking about a continuous random variable here. For continuous random variables we assign probability densities, not probabilities to all the points in the space. The only way to determine probabilities is to integrate the probability density function over an integrable subset of the event space (typically an interval). Observing that the probability of landing on a single event in the space is zero is as trivial mathematically as equivalent to observing that the definite integral from point a to point a of function f is zero. It’s senseless, IMO, to try to imbue this technicality with any deeper meaning.

      In practical terms, measurements are not carried out to infinite precision. As Richard says, there’s a margin of error which means a measurement or observation corresponds to an interval in the event space, not a single point, which we can integrate over that margin to get a nonzero probability for our observation. So no human observation ever corresponds to an exactly zero probability,

      I haven’t yet had the pleasure of reading Richard’s book, I suspect that the relevant event spaces are discrete rather than continuous. So none of that discussion from the lecture would even apply, whether you agree with me or not that it’s a less than ideal presentation of the material.

      For a discrete random variable, we do assign probabilities (technically a probability mass) to each event and use summation rather than integration to compute probabilities. But a countably infinite number of events can each have a nonzero probability mass. For example, the countably infinite variable “the number of times I must flip a fair coin before it comes up Heads” has probability mass function (1/2)^i for every positive integer i. So every single positive integer has a nonzero probability, but all the probabilities for the entire countably infinite set add to one, as is required.

      (Richard, as an aside, the standard treatment of probabilities and probability density functions avoids infinitesimals, I hope this suffices to demonstrate why they are not necessary to dispense with this objection).

  10. abcxyz says

    Off-topic: Richard, would you agree that the NASB is generally the most accurate English translation of the bible?

    • says

      To be honest, they all suck. So this is like asking whose turd smells nicest.

      Every translation has problems. Some are admittedly worse than others. The NASB will do for a start. I use the ASV as my first go-to, but that’s because I know to always check it against others or the original language (e.g. it was updated twice, now as the NRSV). I wouldn’t endorse simply trusting the ASV. But it has certain advantages (e.g. it was composed ecumenically a century before contemporary debates and thus is often more honest, inadvertently).

      Here is a guide to the translation options.

      For problems with the NASB in particular, some examples here.

      Adding to this is that many passages have variants, and often we don’t know which variant was original, and often translations don’t tell you this (that they sometimes do can be misleading, since you may be led to assume they are meticulous about this).

      And so on.

      So, don’t rely on any single translation. Compare several. And when they significantly disagree, be concerned.

  11. abcxyz says

    Richard, what percentage of New Testament scholars would you estimate are believing Christians? I have read that the figure is somewhere above 90%.

    • says

      I don’t know how one would determine that. I’d love to hear about a source (a study? a poll?). 90% is reasonably possible. But I can’t affirm it without evidence.

      One would also like to know the ratio of conservative to liberal, e.g. how many are biblical literalists and/or inerrantists? (Their opinions should not even count toward any authoritative consensus.)

      But then, while we’re at it, one would also like to know the ratio of historicity agnostics there already are among the remaining, when allowed to admit to such anonymously. People might be surprised at the number that resulted.

      Remember, even some devout Christians are willing to admit Jesus didn’t exist (e.g. Thomas Brodie). Although they get drummed out of their jobs when they do. And such a conflict with expectation can drive even motivated reasoning, so that even when allowed to say so anonymously, many might resist admitting agnosticism for fear of what that would mean to their faith, not just their careers.

      But this is all speculation. Plausible. But still.

      Without data, I don’t know what more can be said. (I know David Fitzgerald is trying to pull some data together, but he’s having a hard time of it, because it’s a laborious and vexed business.)

  12. messing says

    Dear Dr. Carrier:

    Actually, we are talking about your complete misuse of “Bayes’ Theorem” in that you conflate it with Bayesianism and that your only proof of your methods relies upon both a misunderstanding of the method you are such a proponent for (despite abandoning it completely in your doctoral thesis) escapes you. I’ve read the works you cite in your both of your books, and while I acknowledge your expertise compared to my amateur familiarity when it comes to ancient history, this is not so of statistics, probability, and BT or Bayesian inference. Here you systematically misrepresent the axioms of probability in fairly basic ways and inconsistent with your own sources. If you wish, I will take you step-by-step through Poupulis, Jaynes, Jeffrerys, Bovens & Hartmann, Urbach, and other vital sources you failed to cite as much as you failed to accurately represent the views of those you did. I couldn’t care less whether or not historical evidence reveals Jesus existed or that Jesus was the patron god of man-boy sexual pedagogy. However, misuse of probability and statistics within many of the sciences are rampant. Now it appears other specialists seem to share this misuse. I left my academic career behind in order to work consulting researchers who suffered from such inadequate training and education. That matters to me.

    Nothing in either of your works, much to my disappointment, presents anything other than a novice understanding of probability theory applied to a philosophical and epistemological understanding of probability. This can be seen clearly when you think you can dismiss the best statistical minds since Bayes’ and the debate between frequentists and Bayesians (and conflate subjective and objective Bayesian interpretation of probability in the process) with a few pages and next to nothing in reference to the vast literature on the subject. Simply put:

    You rely on the assumptions of BT: that hypotheses be collectively exhaustive and mutually exclusive: You fail to meet these requirements utterly. Your sources in PH don’t even advocate the epistemological approach or philosophical interpretation of probability you do. The closest you come to a “formal proof” isn’t formal, proves nothing, and fails to make about the most basic distinction there is in proof theory/formal logic. Your treatment of “BT” is worse thanks to your conflation of an obvious truism unusable for you and the Bayesianism you seek to use but don’t describe (but do relegate to footnotes you are apparently didn’t read.

    • says

      You haven’t identified any actual misuse of Bayes’ Theorem. I actually do not “fail to meet these requirements utterly” but in fact meticulously address and meet them, in the very text you are referring to. My formal proof is by every standard or syllogistic reasoning properly formal (and it is both valid and sound), and indeed proves what it aims to prove (which you have yet to even explain…thus revealing maybe you can’t even articulate what my argument is), and does not have to prove any more than that (so what you mean by “the most basic distinction there is in proof theory/formal logic” is irrelevant to it).

      Even in the hundreds of words you just wrote all you do is repeat the same sentences over and over again in different words. Not once do you identify any actual error in anything I have argued in PH or OHJ. Indeed, so far, one of your claims is a repeated lie (that I didn’t address the the full range of logically possible hypotheses in my construction). The other is unintelligible (you keep claiming there is a flaw in a formal syllogism of mine, yet never identify where it is). And the last is irrelevant (you seem to expect me to have to prove things that in fact I don’t).

      Ordinarily we call that trolling.

      P.S. I was not a Bayesian until after I completed my dissertation (BT was something I studied upon beginning the research project I started after my Ph.D.). So why you think my dissertation should have been Bayesian is beyond me. You must expect me to have a tardis.

  13. Starr says

    This brings to mind a thought I had regarding the association I’ve seen many people make of atheism and mythicism, as though atheists who promote the mythicist position do so out of some need for Jesus not to have existed. But that is absurd, it doesn’t matter if there was a Jesus or there wasn’t, since even if there was it does not mean the supernatural claims about him are true.

    I would think that mythicism should be much more attractive for Christians, not atheists. As modern science and historical research keep hacking away at the truth claims made by the Bible, showing them to be false, most Christians, who choose to remain so, bury their heads in the sand. But mythicism offers them an alternative: reinterpret the basis of their religion as purely cosmic, with all the biblical stories being allegories. Mythicist scholarship gives them a boost, after all now they are just rediscovering the original form of Christianity, lost for ages!

    • says

      That makes too much sense, of course.

      The problem is that it would be a fatal retreat–admitting they were so wrong about their own religion for so long, and that the man who issued their teachings was only ever seen in visions, which may have been hallucinations no less than in any other religion, would be admitting defeat. It would be so easy to dismiss their religion on that account, as easy as they dismiss every other religion on the same account, that Christianity would begin it’s inevitable slide into non-existence (or rather, existence only as the rarefied occupation of tiny fringe cults). I think most Christian leaders are aware of this. (Some aren’t, and actually think your strategy would work, e.g. Brodie, Spong, Sheehan, etc.)

      Which is one reason why arguing for mythicism is not a winning strategy for defeating Christianity. It’s much easier and more effective to point out how ridiculous their current religion already is. They will only entertain mythicism after they’ve admitted their religion that replaced it is ridiculous.

  14. John MacDonald says

    Now that the first round of reviews and comments are over, have you changed your mind at all regarding how likely it is that Jesus existed?

  15. messing says

    I read your comment concerning infinitesimals, which mistakes what I intended (deliberately?). I’ve written about the relative merits of using infinitesimals (rigorously defined at least since Anderson) elsewhere, but if you understood Keisler (or were familiar with the past century of the ongoing debate concerning infinitesimals from Russell to Todorov and beyond), you’d know that axiomatic probability theory has no parallel to neoclassical or non-standard analysis such that continuous probability can be understood as in anyway entailing our inability to partition a continuous probability space as you describe. The link to the lecture on probability makes this clear, as does every single intro probability textbook I’ve come across. It uses a geometric formulation of probability space in terms of points given a continuous function defined for all values (x,y) in the unit square. It is just as easy, however, to consider possible values defined on the interval [0,1].

    Not only are infinitesimals different from zero (there are in fact infinite cardinalities of infinitesimals, e.g. some are larger or smaller than others, and there is a whole system of arithmetic for this, and zero is only one of these numbers: for resources see my comment here),

    1) Your comment refers to a basic site intended for undergrads on probability, and still shows you are wrong:

    For continuous probability, however, events may have probability zero without being impossible, and calculus is needed to sum over these infinitesimals to get a finite number

    The above is a reference to integration and measure theory, but more importantly reveals that we can’t sum continuous probabilities (otherwise, we wouldn’t need integration theory). Hence the rather ridiculous nature of this statement:

    Both points combine: by the laws of calculus, that 90% space is actually the sum of infinitely many theories (all the variants that still share the same general attributes).

    To get this back to basic yet defensible mathematics, consider the probability for a single value/outcome given a continuous probability function. All such outcomes are 0, as all probabilities are expressed as ratios or fractions of the form 1/x and the limit of any particular x value given such a function defined over (0,1) is 0.
    2) Probability spaces, discrete or no, are not made up of theories. Many statisticians, scientists, and applied mathematicians (among others) refer to hypothesis spaces. In machine learning, this means something drastically different then what most would think, yet it is here that bayesianism thrives (because Bayesian inference is ideal for formal models of learning and decision theory).
    3) There are not “infinite cardinalities of infinitesimals”. Infinitesimals are either taken to be the formalized modern version of the intuitive yet debated notion dating back centuries, in which case they no more have a cardinality than do the number 3, the fraction 3/5, or pi, or the term is used informally in which the same holds true. There are infinitely many cardinalities of infinities, or more precisely there are infinitely many infinite sets with differing cardinalities (the smallest being any infinite set with a cardinality equal to that of the integers, and the existence of any infinite set with a cardinality greater than the set of integers but less than that of the reals being undecidable).

    n.b. “arguments must be logically valid, and factually sound” Proving History, p. 45.

    This is what I mean by a failure to distinguish. Soundness is a logical property, making your contrast between “logically valid” vs. “factually sound” confusing. A logically sound argument is necessarily factual. Likewise your mention on the same page of BT being “formally proved, both mathematically and logically, so we know its conclusions are true- if its premises are true.” Apart from the redundant use of formal, mathematical, and logical, a valid derivation isn’t a “proof” and there can be know “proof” that isn’t sound. Proofs depend upon axiomatic foundations, unless we use the term informally. This is akin to your confusing phrasing in your statement (p. 106) that “BT is a logically proven theorem” (are there illogically proven theorems?) and the irrelevant and unsound conclusion that “no argument is valid which contradicts BT”.

    Which is relevant to that proof how?

    Because it proves the equivalent of the following:
    P1: Jesus existed.
    P2: If Jesus existed, BT is a religious belief with no logical foundation.
    C1: BT is a religious belief with no logical foundation.

    The above derivation is obviously and totally wrong. It is, however, valid. Your proof asserts that “no argument is valid which contradicts a logically proven theorem”. This is absurdly incorrect. However, as it is a premise, the conclusion you make follows from it (and P1 on p. 16) and your argument is valid. It’s just wrong. The point is that if nuances like validity are not adequately described such that you spend pages and I can’t understand the pages devoted to a “proof” that is as logically valid as my above (obviously unsound) “argument” about Jesus and BT.

    Funny how that is exactly what I say as well

    You don’t.

    The fact that infinitesimals sum to nonzero numbers was discovered and used even by Archimedes 2200 years ago

    The fact that infinite series can converge as discovered and shown by Archimedes (albeit without a rigorous foundation) says nothing about infinitesimals unless you use the term informally to mean something like “vanishingly small” such as that limit of the function f(x)=1/x as x approaches 0. Infinitesimals lacked rigorous definitions until Anderson’s work in the past ~50 years, and despite this modern analysis still almost always relies on the epsilon-delta formulation of limits rather than infinitesimals. Infinitesimals are not the same as infinitely small “units” (although I admit I was using the term that way; perhaps I should have used the ε,δ-definition of limits instead).

    This is why we work always with margins of error (so we never say or assume the probability is x, but between x and y, and therefore we can never say it’s zero, because when x is zero, y is necessarily nonzero).

    1) “We” don’t always work with margins of error. And even when “we” do, we can still say the probability is 0. This seems to be another mistaking of the properties of continuous variables for discrete, although as you’d be wrong in the continuous case too it’s hard to tell.
    2) Confidence intervals, margins of error, and other similar notions used in probability and statistics are generally used for reasons relating to methodology, not formal systems and thus not probability theory. Even were we to violate basic logic as QM does, and state that a quantum system has ontologically classical observable “values” that are exclusive (such as the location of a particle in one space and another), we can still say that the probability of locating this probability outside of the universe is 0. Likewise, the probability of rolling a 15 given a roll of standard (fair) dice is 0.
    3) Probabilities are defined over a closed interval, meaning that the endpoint 0 necessarily exists for any probability function (this is true even in the discrete case).

    Note that this debate is more smoke and mirrors than anything substantial

    Which you “show” by conflating terms, misrepresenting your own sources, and making claims like your resolution of the ongoing dispute over frequentists vs. Bayesian interpretations made by the greatest statistical and mathematical minds of the last century (including some you cite) which you “demonstrate” are groundless without recourse to virtually any of the literature, no mathematical argument, and clearly incorrect statements about what Bayesian and frequentist interpretations are. Whether you understand the nuances here is irrelevant- when you can’t even be bothered to use the terms “formal” and “valid’ in their technical sense, how can we trust the incredibly simplified and thus hugely inaccurate summation of decades of highly technical debate in the few pages you so “solve” the problem, still less accurately characterize how what every opponent of Bayesianism agrees is sound is in fact equivalent to what they define themselves as in contradiction to?

    • says

      1) Your comment refers to a basic site intended for undergrads on probability, and still shows you are wrong:

      No, it doesn’t. You have yet to show that you even know what I was saying, much less in what way it is wrong, even less how that source says so.

      …we can’t sum continuous probabilities (otherwise, we wouldn’t need integration theory).

      Isn’t integration theory specifically for the summing of an infinite series? It allows you to do this by adding them as defining areas–which would include probabilities when defined as divisions of an area–which is what a frequency is. Again, Archimedes did this. Newton did this. It’s not even new.

      So you aren’t making a lot of sense here.

      Nor can I discern your point.

      Are you claiming probability reasoning is useless because we can’t account for infinitely many hypotheses, therefore we can’t account for any?

      That sounds like radical skepticism to me. Which would entail agnosticism about the historicity of Jesus.

      Hence the rather ridiculous nature of this statement:

      Both points combine: by the laws of calculus, that 90% space is actually the sum of infinitely many theories (all the variants that still share the same general attributes).

      To get this back to basic yet defensible mathematics, consider the probability for a single value/outcome given a continuous probability function…

      Okay, I’m starting to think you are a troll.

      Because what you go on with amounts to an assertion that all calculus is false.

      So either you are a terrible communicator, or…what?

      We are not talking about “single outcomes,” we are talking about whole collections of possible outcomes. When we say a theory has a 90% probability of being true, we are saying that of all theories which rest on evidence of that kind, 9 out of 10 of them will be true. Not just this one single theory. All logically possible theories that rest on an equivalent scale of evidence.

      And “a theory” is not a unique entity. There are infinitely many theories it can be subdivided into. For any theory H, there is theory H+A, H+A+B, H+A+B+C…ad infinitum (more properly H+A(~B~C…etc.) and H+A+B(~C…etc.)). The sum of the prior probabilities of all those theories will equal the prior probability that simply H. Conversely, “the prior probability that H” can thus be divided into infinitely many theories, each with (as a result) an infinitesimal prior. Neither fact has any effect on the prior for simply H being 90%.

      If you are gainsaying this, then you are the odd man out.

      2) Probability spaces, discrete or no, are not made up of theories.

      They can be made up of anything. People. Cats. Murders. Orgasms. And theories.

      If it has a frequency, it occupies a probability space.

      3) There are not “infinite cardinalities of infinitesimals”.

      I assume what you mean to say is that technically each infinitesimal corresponds to a cardinality of infinity, as being the converse thereof. But that’s just a semantic pedantry of reference, not a substantive objection to anything I said.

      n.b. “arguments must be logically valid, and factually sound” Proving History, p. 45.

      This is what I mean by a failure to distinguish. Soundness is a logical property, making your contrast between “logically valid” vs. “factually sound” confusing.

      How so? You have yet to identify a single instance in which I confuse them.

      Instead, you cite evidence of my distinguishing them, as evidence I didn’t. Which makes no sense.

      You even did it again…

      A logically sound argument is necessarily factual. Likewise your mention on the same page of BT being “formally proved, both mathematically and logically, so we know its conclusions are true- if its premises are true.”

      Here I clearly distinguish between validity, and soundness. And I correctly distinguish one as a logical property, and the other as a factual one.

      And your objection is?

      Apart from the redundant use of formal, mathematical, and logical, a valid derivation isn’t a “proof” and there can be know “proof” that isn’t sound.

      Yeah. We all agree on that. What’s your point?

      In what way is my proof on pp. 106ff. unsound? (I am assuming you agree it is valid.)

      That requires you to point to a premise in it that is false or not probably true.

      So, which premise in that syllogism are you saying is false or not probably true?

      (And< BTW, what would it's being so have to do with my not knowing the difference between validity and soundness?)

      “BT is a logically proven theorem” (are there illogically proven theorems?)

      There are theorems that aren’t proven. And there are theories (lay readers don’t know the difference) that have not been proven by deductive logic, but inductive data analysis.

      So is your problem the fact that I am violating the Law of Ivory Tower Elitism by using language that can be understood by ordinary people?

      What has my chosen dialect have to do with what it is being used to say?

      …and the irrelevant and unsound conclusion that “no argument is valid which contradicts BT”.

      How is it unsound? You have yet to explain this.

      Are you actually saying things can be simultaneously logically necessary and false?

      You would seem then to be one of the most radical skeptics I’ve ever met. Denying even that logic is true!

      Because it proves the equivalent of the following:
      P1: Jesus existed.
      P2: If Jesus existed, BT is a religious belief with no logical foundation.
      C1: BT is a religious belief with no logical foundation.

      WTF?

      I have no idea what this bizarre syllogism has to do with anything in either of my books.

      You have yet to show anything in my syllogism is false or not probably true. So this isn’t even usable as an argument from analogy–since we can easily show P2 to be false. So let’s see you do that with any of my actual premises.

      Unless you are claiming my P1 (p. 106) has the same probability of being true as your P2…and good gods almighty, is that what you are seriously claiming!?

      1) “We” don’t always work with margins of error.

      “We” damned well better have–in all matters empirical. Because if we don’t, then we’ll be looking at unsound conclusions across the board.

      It’s strange to see someone so obsessed with the importance of logical soundness, advocating a practice that guarantees a lack of logical soundness.

      And even when “we” do, we can still say the probability is 0.

      You seem to be confusing “is” with “could be.” If there is a margin of error (your own statement “even when we do” entails you are granting this), then it cannot be the case that the probability “is” zero. Because then there would be no error margin attached to that statement. If there is an error margin, then it has to be somewhere above zero. Otherwise there is no margin of error being stated at all. That the value could be zero is not at issue. What is at issue is that we only know it is somewhere in the margin, not that it is indeed zero. (The probability of it being zero can be so high that we proceed as if it is, but that’s not the same thing.)

      I am not the one who is confusing “could be” with “is” here. That’s on you.

      Which you “show” by conflating terms, misrepresenting your own sources, and making claims like your resolution of the ongoing dispute over frequentists vs. Bayesian interpretations made by the greatest statistical and mathematical minds of the last century (including some you cite) which you “demonstrate” are groundless without recourse to virtually any of the literature, no mathematical argument, and clearly incorrect statements about what Bayesian and frequentist interpretations are.

      This is all false. You have not shown a single instance in which I have actually misrepresented any source, you have not shown a single instance in which I have conflated any terms, you have not shown a single instance in which my resolution of the frequentist-subjectivist debate even contradicts anyone (much less the people you claim to revere), you have not shown a single instance in which “any of the literature” refutes me on that, you have not shown a single instance in which I have misstated “what Bayesian and frequentist interpretations are” (or at least sometimes are, since they vary), and I actually present a considerable amount of mathematical argument in the section you are referring to.

      All of this, after all this time, is convincing me that you are a troll, and that you don’t really have any legitimate objections to my work.

  16. says

    I’ve ordered the book — I haven’t gotten it yet — but I’m already wondering how apologists will respond to your Bayesian analysis. They’ll probably fudge things massively in their favor, but on the off chance that they have legitimate disagreement, the way to reconcile that using the framework of BT is to combine likelihood ratios.

    • says

      To be fair, I’m not really very interested in apologists’ opinions anymore. They are guaranteed to be bullshit even before the word go. But if you see any such, to which you can apply your idea, definitely do that, either here or elsewhere and link to it here. I’d like to see how something like that would work (esp. given how hard it will be to figure out what an apologist is even arguing so as to quantify it or apply their quantity in the right place).

  17. messing says

    Isn’t integration theory specifically for the summing of an infinite series?

    No. If your familiarity with undergraduate mathematics fails here, then you have something of a problem when it comes to probability theory and Bayesian analysis. First, for most of the period from Newton and Leibniz there was not even a good informal conception of series, but integration was defined by antidifferentiation. Second, while series are key in analysis (i.e., the area of mathematics of which calculus is the “first step”), it wasn’t really until measure theory and topology that integration theory came into its own, and neither depend on elementary notions such as “summation”. Third, and most obviously, were integration a form of summation we could express it using any method we do to express summation (although not necessarily vice versa). We can’t, because it isn’t, and more importantly the basic integration you took isn’t really integration (there’s a great free textbook you can check out on this, both integration theory for the amateur and the divergence between undergraduate calculus and modern integration:

    “For all of the 18th century and a good bit of the 19th century integration theory…was simply the subject of antidifferentiation. Thus what we would call the fundamental theorem of the calculus would have been considered a tautology: that is how an integral is defined…This is often expressed by modern analysts…by the Newton integral…

    The technical justification for this definition of the Newton integral is nothing more than the mean-value theorem of the calculus. Thus it is ideally suited for teaching integration theory to beginning students of the calculus…For these reasons we have called it the calculus integral. But none of us teach the calculus integral. Instead we teach the Riemann integral. Then, when the necessity of integrating unbounded functions arise, we teach the improper Riemann integral. When the student is more advanced we sheepishly let them know that the integration theory that they have learned is just a moldy 19th century concept that was replaced in all serious studies a full century ago.
    We do not apologize for the fact that we have misled them; indeed we likely will not even mention the fact that the improper Riemann integral and the Lebesgue integral are quite distinct…We also do not point out just how awkward and misleading the Riemann theory is: we just drop the subject entirely.”

    (italics in original; emphases added; see here: http://classicalrealanalysis.info/documents/T-CalculusIntegral-AllChapters-Portrait.pdf)

    As much of integration cannot be formulated in terms of the summation, and has surprisingly little to do with the integration concepts you learned in your calculus course, it is not surprising that you conclude as you do. The same cannot be said for you comments about infinitesimals.

    You have yet to show that you even know what I was saying, much less in what way it is wrong, even less how that source says so

    Fair enough. I maintain that most of your presentation of probability theory, inference, formal logic, and mathematics is either simplistic to the point of distortion or simply wrong. In terms of BT, this is clear and obvious. You do not ever work with a probability space that is both discrete, collectively exhaustive, and mutually exclusive. Your presentation of hypotheses in a book I have been looking forward to for years before Proving History misuses formal notation to denote a central hypothesis and its negation, but the notations is flawed. The use of formal notation for negation entails formal negation, but you don’t formally negate your hypothesis with the proposition you assert is the negation of h-bar or h-prime or whatever term you wish. Thus, your partition of your probability/hypothesis space is logically flawed from the start. It is not collectively exhaustive and thus your use of (your idiomatic formulation of) BT is likewise flawed.

    I wished to show something of actual Bayesian theory and probability by looking at what probability distributions, probability spaces, and real partitions of probability spaces consist of. In order to do so, this requires an understanding of both summation (in the case of discrete probabilities) and integration/measure theory (for continuous). Your references to infinitesimals, which have no rigorous (useable) formulation prior to the 60s (yet you refer to Archimedes) and before that were simply equally inadequate foundations for any probability theory and analysis in general as were limits prior to the modern, standard formulation. For a real treatment of infinitesimals in the context of probability theory, see e.g., Barrett, M. (2010). The possibility of infinitesimal chances in E. Eells & J. H. Fetzer (eds.) The Place of Probability in Science: In Honor of Ellery Eells (pp. 65-79) (Boston Studies in the Philosophy of Science Vol. 284). Springer. As for infinitesimals, the issue is much more nuanced then can be presented as the informal notion was used for centuries before it had to be abandoned for a thoroughly unintuitive alternative because the greatest mathematical minds couldn’t provide a rigorous foundation for analysis using it. You reference one of the few standard texts on infinitesimals, but then relate these ideas to limits and integrals by confusing all three.

    Again, Archimedes did this. Newton did this. It’s not even new.

    “The notion of a limit is complicated in itself; historically, coming up with a correct definition took 200 years. It wasn’t until Karl Weierstrass [sic] (1815-1897) that calculus became a rigorous subject.”
    p. 88 of Hubbard & Hubbard’s Vector Calculus, Linear Algebra, and Differential Forms.

    About the kindness one can say of Archimedes is

    In his To Erathostenes: Method on mechanical theorems in Archimedes presents us with arguments that are not flawless undisputable proofs, but have a strong heuristic power

    Lolli, G. (2012). Infinitesimals and infinites in the history of mathematics: A brief survey. Applied Mathematics and computation, 218 (16), 7979-7988.

    And just to reiterate:
    “Despite its power and beauty the foundations of infinitesimal calculus were shaky. The glaring contradictions were apparent to all and the subject of severe criticism…Very many attempts were made to free foundations of infinitesimals from contradiction both by Leibniz and later…Leibniz and his followers were never able to state with sufficient precision just what rules were supposed to govern their new system including infinitely small as well as infinitely large quantities.”
    From Stroyan and Luxemburg’s Introduction to the Theory of Infinitesimals.

    The method of exhaustion Archimedes used was neither rigorous nor accurate- had it been, then he could have extended it from the functions he neither understood, defined, nor was capable of extending to functions mindless high-school AP calculus students can. This takes nothing away from his genius, of course, simply from the ridiculous notion that one can look to my statement about infinitesimal probabilities in the context of intro calculus and ancient Greek geometry.

    Nor can I discern your point.

    Are you claiming probability reasoning is useless because we can’t account for infinitely many hypotheses, therefore we can’t account for any?

    Not at all. I’m claiming that your use of “BT” is neither Bayes’ theorem, Bayesian inference, nor accords with probability theory. You violate basic formal logic, you fail to adequately partition a probability space that you can’t even in theory hope to yet also fail to use statistical/probabilistic analyses designed to avoid such requirements. You present a “proof” that means nothing because it is wrong from the start and your attempt to “partition” your probability space contradicts both formal logic and probability theory. In short, by failing to formally define a real probability space, you can’t possibly partition it; and you cannot make any logical conclusions based upon your failure to use the methods you refer to in Proving History (which are not using Bayes’ theorem but subjective probability & inference you mischaracterize when you attempt to conflate it with the frequentist view, as though your understanding of statistical inference were so far above others that in a few pages bereft of mathematical arguments or references you defeat at least a century of arguments among actual mathematicians). I would love to give you citations here, but the problem is that you cite the sources that vehemently disagree with you. I am not certain whether this is because you failed to understand your own sources (which is understandable; you don’t have a background in mathematics, science, statistics, or modern philosophy of probability) or because you didn’t read them carefully enough. Thus, I’m not sure how to proceed to show precisely how you present BT proper that you cannot use, fail to present the Bayesianism your sources do, and violate the logic underlying what you purport to be your basic approach.

    We are not talking about “single outcomes,” we are talking about whole collections of possible outcomes. When we say a theory has a 90% probability of being true

    We don’t. It’s a misuse of the “theory”, probability, and epistemic justification. However, my point was that when you do not formally define your probability space you cannot justify the use of Bayesian inference which relies on probability distributions you reject in favor of ad hoc best guesses based not on priors but on your analysis of the evidence s.t. you have no basis for posteriors (certainly not given that your formulation of Bayesian inference is the basic extension of conditional probabilities). The essence of the Bayesian method you use is supposed to consist of representing your uncertainty based upon probability distributions that make-up (mostly) your current beliefs or epistemic probability. You then update these beliefs using these distributions s.t. the given your new information your beliefs chance according to these prior distributions. You don’t have new information, and your mathematical model isn’t Bayesian but BT, which requires an very different approach. It’s basically worthless unless you can partition the probability space exactly in a way that would make your use of it pointless because you do not ever gain new information. This is what I mean by conflation of BT and Bayesian inference.

    How is it unsound? You have yet to explain this…WTF? I have no idea what this bizarre syllogism has to do with anything in either of my books.

    This is basic formal logic. The premise/proposition that no argument is valid that contradicts BT, is clearly and obviously incorrect. It cannot be used in any derivation that may be deemed sound. This is because validity need not consist of true premises. Ergo, any argument that contradicts BT but that does so such that were its premises true, it’s conclusions would be true as well, is a valid argument. The “proof” I gave was valid and contradicts BT. It is not sound, but neither is your “proof”, making it rather pointless. Admittedly, to the layperson it seems logical and is only flawed for technical reasons. But your book was intended to justify your formal approach and fails to make basic distinctions required of formal logic, thus rendering an extended proof nothing more sound than “if the moon is made of green cheese, than BT is false. Ergo, BT is false”. You include a false premise, and as you yourself state an argument is only as strong as its weakest link: infinitely many ridiculously unsound but valid arguments can contradict any proven theorem.

    I’m tired, sleep-deprived, and just realizing that if I go on I will simply make less sense. I invite you to carefully read what I have said (typos and grammatical errors do to not proof-reading aside) and whether you think what I have said is insane idiocy or not, ASK WHAT I MEAN AND WHAT MY BASIS IS AND WHERE IT IS THAT YOU DO NOT UNDERSTAND. You do not have background in mathematics for many of the arguments you make here and in your books, and your misrepresentation (I do not believe it deliberate) shows this.

    Believe it or not, I was a huge fan of yours. Perhaps I am overly critical because of this. But I am very tired of being dismissed by a historian who dismisses mathematicians, scientists, and even philosophers who disagree with him about their fields and simultaneously derides those who write historiography but lack credentials in history. I am rather thoroughly fed up with being insulted by someone who doesn’t understand fairly basic material I’m trying to explain because no matter how poorly I do so, you shouldn’t require any explanations. Maybe being a thoroughly pissed off fan makes me a troll, but there is nothing I’ve said which was not either intended to be an explanation of simply a statement of the basics or both.

    • says

      Your argument has descended into the almost unintelligible at this point.

      This infinitesimal nonsense is a huge distraction. I can’t even fathom why you think it’s relevant. But if infinitesimal areas can be summed (and they can), infinitesimal probabilities can be summed, as they are areas (partitions of a whole). So either you are saying infinitesimal areas cannot be summed (and therefore calculus is bogus), or you are saying something that has nothing to do with my use of probabilities.

      But back to something that sounds at least remotely relevant:

      1. You have still not explained where I “violate basic formal logic.” Assertion is not argument. This has been several times now, several requests for you to identify where there is a logical error, and you still have not identified this supposed error. I have repeatedly asked you to identify the step in the syllogism that is invalid, or the premise that is false or not probably true. You have repeatedly dodged that question and instead just repeat the assertion that you found one. But apparently only Schrödinger’s cat knows where it is. I should remind you that the purpose of stating an argument in a formal syllogism is so critics can do this. Your refusal to do it is therefore a sign of contempt for rational discourse.

      2. I have already explained how I “adequately partition [this] probability space,” and you keep ignoring me, and the book. Again you just keep repeating the assertion that I didn’t. An assertion anyone who reads the book can see is false. So I don’t know what you think you are trying to pull here.

      3. I don’t know what you are expecting by the “use [of] statistical/probabilistic analyses.” All I claim to do is quantify my estimates of odds and calculate what the consequences are of accepting those estimates. You have yet to explain what is illegitimate about that.

      4. You have also not identified how my actual “partitioning” of the probability space “contradicts both formal logic and probability theory.” You have yet to even quote or describe my partition method. Again, if it is illogical, show the actual step (in the actual book) where I fail to maintain the excluded middle, or where I incorrectly assign a probability to one of the partitions. Don’t just keep asserting that I did. Somewhere. Somewhere again apparently only Schrödinger’s cat knows where. Assertion is not argument. Need I refer you to the Monty Python skit?

      5. You claim I fail “to formally define a real probability space,” which is obscure in its meaning. Are you claiming a possibility space is not real and therefore can’t be partitioned? Because that would entail no hypothesis (even in science) has any knowable probability of being true (since alternative explanations of any scientific observation are infinite and thus divide up the whole possibility space). So I am hoping that is not what you mean.

      6. You claim Bayes’ Theorem is not “subjective probability & inference,” which is weird because every other mathematician I know is happy to use Bayes’ Theorem in precisely that way, and I even cite several of them. For it is a theorem for solving inverse probability problems, and drawing logically valid inferences from subjective estimates of frequency is an inverse probability problem. You then gainsay my demonstration that subjective probabilities are really estimates of error frequencies, yet you don’t at all engage with my demonstration or even show any awareness of what it actually is. Much less what’s wrong with it.

      7. I ask for citations of these experts you claim refute me, and you provide none. You don’t point out where they disagree with me, or even what they disagree with. You just insist they do. Somewhere. Somehow. Only Schrödinger’s cat knows.

      These failings of yours, your refusal to even discuss what my actual arguments are, and your assertions that they contain flaws but continued refusal to identify where those flaws are, is starting to look insane.

  18. messing says

    I begin with 1) not because it is the most important by any means, but because it speaks to a far more general, pervasive problem- the tendency to make claims involving technical terms simplistically, informally, misleadingly, and/or inaccurately. Why you do not, in fact, offer a “formal proof” using “formal logic” and why your argument is wrong simply because you misunderstand or misrepresent the basics of formal logic is just the best example. I then address 2), but as this relates to several other objections you have, I will temporarily leave my response to these as my response to 2). I will address your other problems in more detail in a separate reply lest I give too much detail at once.

    1) The main problem with your “proof” in Proving History (PH) (pp. 106-109) is, again, that it is an argument which, while valid, is not sound, and therefore doesn’t show anything. To see this we have only to look at a book you yourself have recommended: The Philosopher’s Toolkit by Baggini & Fosl. Validity is defined as follows:

    ”Validity is a property of well-formed deductive arguments, which, to recap, are defined as arguments where the conclusion in some sense (actually, hypothetically, etc.) follows from the premises necessarily…The tricky thing, in any case, is that an argument may possess the property of validity even if its premises or its conclusion are not in fact true. Validity, as it turns out, is essentially a property of an argument’s structure. And so, with regard to validity, the content or truth of the statements composing the argument is irrelevant.”

    (p. 13; italics in original)

    Soundness is defined as follows:

    To say an argument is valid, then, is not to say that its conclusion must be accepted as true. The conclusion is established as true only if (1) the argument is valid and (2) the premises are true. This combination of valid argument plus true premises (and therefore a true conclusion) is called approvingly a ‘sound’ argument…If you accept an argument as sound, you are really saying that one must accept its conclusion.

    (p. 15; italics in original)

    Your second premise is “[n]o argument is valid which contradicts a logically proven theorem.” However, as a source you recommend states quite clearly, an argument can be wrong and be valid. Infinitely many valid arguments can contradict any theorem or true statement. I gave an example already but will give an even simpler, more direct valid argument that shows your argument/proof isn’t sound:

    Premise: BT is a logically proven theorem
    Premise: All valid arguments contradict every logically proven theorem.
    Conclusion: All valid arguments contradict BT.

    The fact that the premises are not true is utterly irrelevant: the above argument is valid because were the premises true then they entail the conclusion (i.e., the conclusion necessarily follows from the premises). It isn’t a sound argument because the premises are wrong. However, the second premise in your “proof” is also wrong as I’ve shown by yet another counter-example. Therefore, your argument is not sound and your conclusion is false.

    One might object that, as you use the terms “formal proof” and “formal logic”, not argument, that different rules apply. In fact, this makes you even more wrong. On p. 106 you entitle your section “Formal Proof of Universal Applicability” and you state you will “establish [your] conclusion by formal logic.” In logic and mathematics the term “formal” is used for a reason:

    “A formal proof is a proof written in a precise artificial language that admits only a fixed repertoire of stylized steps.”

    Formal Proof- Theory and Practice

    Not only do formal proofs use formal language (i.e., symbols), but every step is justified within a formal system:

    “A formal proof…is a sequence of assertions, the last of which is the theorem that is proved and each of which is either an axiom or the result of applying a rule of inference to previous formulas in the sequence; the rules of inference are so evident that the verification of the proof can be done by means of a mechanical procedure. Such a formal proof can be expressed in first-order set-theoretical language”

    p. 15 of Tall et al.’s (2012) Cognitive Development of Proof in G. Hanna & M. de Villiers (Eds.) Proof and Proving in Mathematics Education (New ICMI Study Series Vol. 15). Springer.

    Finally, while “soundness” is defined differently, the difference only renders more clearly that you do not, in fact, give a formal proof at all:

    ”A proof system consists of a set of basic rules for derivations. These rules allow us to deduce formulas from sets of formulas. It may take several steps to derive a given formula G from a set of formulas F, where each “step” is an application of one of the basic rules. The list of these steps forms a formal proof of G from F.
    Of particular interest is the relationship between the notion of formal proof and the notion of consequence. We want a proof system that is sound. That is, we want the following property to hold.
    (Soundness) If a formula G can be derived from a set of formulas F. then G is a consequence of F.”

    p. 13 of Hedman, s. (2006) A First Course in Logic: An Introduction to Model Theory, Proof Theory, Computability and Complexity (Oxford Texts in Logic)

    You have good reasons for not using formal logic or giving a formal proof, but this just illustrates how far from providing one you are:

    ”To nonmathematicians, formal proofs can be intimidating and sometimes incomprehensible. Rucker (1982, p. 274) illustrates how cumbersome the process of writing out a formal proof can be with an example of such a proof of (∀y) [0 + y = y], that is, for all y, 0 + y = y. The proof takes 17 steps and uses on the order of—I am estimating—400 to 500 symbols. Rucker argues that, despite their ‘nitpicking, obsessive quality,’ fully formalized proofs ‘are satisfyingly solid and self-explanatory. Nothing is left to the imagination, and the validity of a formal proof can be checked simply by looking at the patterns of symbols. Given the basic symbols, the rules of term and formula formation, the axioms and axiom schemas, and the rules of inference, one can check whether or not a sequence of strings of symbols is a proof in a wholly mechanical fashion’”

    p. 105 (emphasis added) of Nickerson, R. S. (2010). Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs. Psychology Press.

    You don’t use “formal logic” and even if you did, the reason for the term “formal” in “formal logic” (or “formal system”, “formal language”, etc.) is form. The reason to reducing arguments to sets of symbols is to highlight their form or structure to better and more easily determine validity. However, validity is system-relative. In my symbolic logic course years ago, we used a formal system which did not permit the justification of any step in a derivation via MT, MP, De Morgan, and a number of other inference rules most systems of propositional and predicate calculus contain. Thus, arguments which are both valid and sound can be formally invalid simply because the argument uses valid inferences within a system that e.g., requires such inference types to be derived rather than used to justify a step in a proof, lemma, derivation, etc.

    You don’t use any formal system, but it doesn’t matter:

    ”The conceptions of validity,…considered thus far, are system-relative and apply only to formal arguments. What is going on, though, when one udges an informal argument to be valid? One is claiming, I take it, that its conclusion follows from its premises, that its premises couldn’t be true and its conclusion false. (If, besides being valid, an argument has true premises—and so, being valid, true conclusion too—it is said to be sound).

    p. 14 (italics in original) of Haack, S. (1978). Philosophy of Logics. Cambridge University Press.

    Informally, your argument is valid but not sound because it mistakes what validity is, thereby asserting something about the nature of valid arguments that is incorrect. It is not true that a valid argument can’t contradict a proven theorem any more than it is true that a valid argument can’t contradict a fact or true statement. Valid arguments can be absurd, obviously wrong, contradict facts, etc., so long as the conclusion follows from the premises. Therefore, every conclusion that follows in part by your 2nd premise “[n]o argument is valid which contradicts a logically proven theorem” is not a sound conclusion and your entire “proof” is wrong. You don’t prove anything (formally, logically, or in any way whatsoever) although you do provide a pretty thorough misuse of terminology.

    Worse still, even if we fixed the terms used, we would still be left with the mischaracterization of BT.

    2) You continually use the notation h & ~h, but formal negation (as indicated by the negation symbol) isn’t simply what you think to be the opposite of some proposition. If h is “Jesus was a historical person mythicized” than ~h is “it is not the case that “Jesus was a historical person mythicized”. The latter proposition is not formally equivalent to “Jesus was a historical person”, it is not formally equivalent with “Jesus as a mythical person historicized”, it is formally equivalent only with syntactical manipulations of some symbolic representation of h and truth-functionally equivalent WFFs. You can’t partition a probability space when you fail to do something so basic as formally defining the hypothesis space such that you have mutually exclusive and collectively exhaustive propositions which make it up. This begins the answer to the false statement below:

    I have already explained how I “adequately partition [this] probability space,” and you keep ignoring me, and the book.

    You refer the reader to Papoulis’ probability text in PH for a “complete proof” of BT. Papoulis provides a simple but clear definition of partitions

    Partitions A partition Uof a set f is a collection of mutually exclusive subsets Ai whose union equals f (Fig. 2-5)
    A1+…+An=f [and] AiAj= ∅

    We get the proof in Papoulis’ text only via such a partition:

    If A= [A1,…,An] is a partition of f and B is an arbitrary event (Fig. 2-5) then
    P(B) = P(B|A1P(A1)+…+P(B|AnP(An)
    [this is followed by the proof of the above equation]…”The result is known as the total probability theorem.

    As Papoulis shows, BT is a consequence of some algebraic manipulations of the above equation together with the equation given by the proof- P(BAi) = P(B | Ai) P(Ai), whence “we obtain Bayes’ Theorem
    P(Ai | B) = {P(B | Ai) P(Ai)}/{ P(B | A1) P(A1)+…+ P(B | An) P(An)}”

    Because I can’t provide a scan of the pages from Papoulis and typing here doesn’t make the math particularly clear, I located an online source after quick search that is quite similar (most texts are at this point, actually), but note that it just so happens the author reverses the uses of B & A, so that the partitions are now B1, B2,etc.
    General Probability, III: Bayes’ Rule. The standard examples given in probability texts for using Bayes’ theorem (for some reason many texts use marbles/balls of different colors taken from urns rather than something more creative or using a less archaic word) aren’t particularly helpful because in the real-world we are typically working with sample spaces with outcomes to which we assign probability measures based upon imperfect knowledge. One frequent example is the use of medical testing and false positives, but only when idealized does this actually yield any probability which Bayes’ theorem proves is correct. BT is really no different than other straightforward rules of probability such as the multiplication rule (if an event can occur in n distinct ways, such as the number of possible sequences given n coin tosses, the probability for any one of them if n! or, for the coin toss, .5^n). More technical definitions of probability spaces require the standard triple (

    • Anonymous Coward says

      The post seems to have been cut off. it ends with an open paren in the middle of a sentence. I can only see point 1 and part of point 2.

      Concerning point 1, you didn’t probably have a chance to read my other post to you yet (under a different post on this blog) but here I’ll say that yes, this is a spot where Carrier seems to be using logical language extremely loosely at best. The thing is, though, the passage you’re taking him to task over could simply be deleted from the book entirely–the argument is not needed, it’s just a kind of “and not only that!” argument, added on for oomph. For that reason, your criticism here does seem to be pedantic (again, as I discussed in that other post which you probably haven’t had a chance to read yet). You’re accurately criticizing points of usage, but concerning matters that don’t seem to have real significance for the overall issue being discussed.

      One might argue that if he goes wrong about this here, he’s lost credibility. But the credibility of the reasoning he’s expressing stands or falls independently of how he expresses it. If he were expressing the argument using complete gobbledygook, then of course we’d have no reason to think he was expressing credible reasoning (even if he, unbeknownst to us, was trying to express arguments that are in fact perfectly cogent). But Carrier’s not writing in gobbledygook. He’s writing in English, and it’s not that hard to figure out what he’s after even when he’s unclear and misspeaks.

      Concerning point 2 only part of it showed up, but do I understand correctly you’re talking about the part in OHJ where he says he’ll treat (IIRC) “Jesus was a historical man mythicized” and “Jesus was a mythical man historicized” as his “h” and “-h”? Are you taking him to task over the fact that these aren’t strictly negations of each other? Carrier said he answered that point in the book, and you said he didn’t, but I clearly recall that he did–he said the other possibilities in the probability space are so miniscule in volume that he will consider it safe to simply treat their size as zero. That leaves just the two possibilities, and it is fair to call them h and -h since that notation isn’t logical notation but probabilistic notation, just meaning “a set of points within the unit interval” and “the complement within the unit interval of that set of points.” But maybe I’m misunderstanding what you’re referring to in the discussion of point 2.

    • messing says

      I don’t know why my post was cut off (perhaps there is a length cut-off, or my use of html?). Here is the rest:
      More technical definitions of probability spaces require the standard triple (

    • says

      That cut off in the same place.

      This means there is some sort of code after the ( that is being rejected by the web server (it will reject anything it suspects of being an attempt to hack the site, so any unrecognized code gets cut).

      Because it happens after a ( I suspect this is because you inserted a string of characters that have coding identities in some browsers immediately after the (. Fixing that requires changing the characters you use, or else boxing the string, i.e. write < code> before the string (angle brackets and all) and code> after it. That will instruct the system to ignore it and just print it as-so. But beware: if the string you are boxing itself contains angle brackets and slashes, you have a redundancy error to workaround (since those will always be read as code, unless broken up by appropriate boxing).

      If that's not it, I can only recommend pasting the text you are trying to insert here into a text-only window (for example, in Mac TextEdit, open a new file and set it to Text Only, then paste in the text you can't get to work here, then copy it back out, and paste that here...all code will then have been stripped clean, and it should work). But that won't fix the problem if the problem is an ASCII character string that has coding implications. The only solution then is to simply change the string, or box it (per above).

    • says

      Your second premise is “[n]o argument is valid which contradicts a logically proven theorem.” However, as a source you recommend states quite clearly, an argument can be wrong and be valid.

      How does that being true invalidate the premise? It is true that any argument that contradicts a logically proven theorem is invalid. That valid arguments can also be false is irrelevant to that statement. The premise does not assert that any argument that does not contradict a logically proven theorem is sound.

      So what is your objection to the premise I actually have in the syllogism?

      I cannot see any intelligible objection here.

      Likewise down the line. You have presented no intelligible objection to any premise in my syllogism. And you just affirmed it is valid. So you can have no sound objection to the syllogism’s conclusion. Unless you wish to deny all of logic itself.

      You continually use the notation h & ~h, but formal negation (as indicated by the negation symbol) isn’t simply what you think to be the opposite of some proposition.

      This is also unintelligible. I only ever use ~h as the negation of h. That’s correct. You give no examples of my doing anything else. In the section in question I carefully explain that I do not simply assume h is “Jesus was a historical person mythicized” or that ~h is “it is not the case that “Jesus was a historical person mythicized”. I correctly demarcate four possibilities and point out two have extremely small priors, so when we fold them in to the remaining two they have no further consequence to the math.

      I am very extremely explicit about this.

      I say that “Jesus was a historical person not mythicized” is included in h. We just don’t have to care that it is, because its priors and consequents are so low that accounting for it further will be invisible mathematically at the resolution we are rounding to. I explain this explicitly and clearly. I also say that “Jesus was a mythical person not historicized” is included in ~h. And the same follows. There are no other possibilities. The entire probability space has been demarcated by h and ~h. No space remains.

      How it is that you still don’t understand this, despite my being so incredibly specific about it in the book, completely bewilders me.

  19. Steve Watson says

    12.1

    “To be honest, they all suck.”

    Then a translation and apparatus that doesn’t is called for. Would you countenance doing, or contributing to doing,such a thing? I would certainly be up to contributing to financing such an endeavour.

    Either in the presence or absence of being able to stand on such shoulders, we should have a knowledge of koine Greek (Not withstanding a large parcel of other skills.) to be able to make a judgement for ourselves. I had cause a couple of months ago to want to check someone’s argument for a textual corruption. I had the devil of a time tracking down the text and apparatus they were using. I mistakenly took it that the translations and working texts would be much of a muchness. When I sampled the texts and tranlations available through Bible Gateway, I could not find anything in the remotest agreement with what I was looking into. It was only the fact that I was taking New Testament Greek and could ask an expert for pointers that I found out the publisher of the text and apparatus had been subsumed (twice) into another body. I was only then able to find what I was looking for and assure myself the argument had not been pulled out of the air.

    If the evidence base and methodologies of our experts are in error, how are mere mortals supposed to proceed?

    • says

      Would you countenance doing, or contributing to doing, such a thing?

      Bible translations take whole teams of highly paid full time experts several years.

      So, uh, no.

      It’s easier to just learn ancient Greek.

      And then buy an Aland at al. and a set of Swansons (although that series has stopped at Galatians; no one has yet taken it back up since Swanson’s death to complete it; yet it remains a required reference for textual variants).

  20. messing says

    Dear Dr. Carrier:

    I struggled to find the clearest, simplest example of a partitioned hypothesis space using a Bayesian analysis. By “clear” I mean not only is the example straightforward but is readily compared to an explanation you provide. Also, I naturally limited myself to academic sources as citing some website is no better than simply giving you an example of my own design. Hopefully, the following suffices.

    In Proving History (PH), you state:

    For example, if testing five hypotheses altogether (yours and for others), then you must ensure that P(h1|b)+P(h2|b)+P(h3|b)+P(h4|b)+P(h4|b)+P(h5|b)= 1…The fact that all priors must sum to one is a useful aid to estimating priors. First you exclude all hypotheses with vanishingly small priors. For example, “space aliens did it” is always so inherently improbable that its prior will surely be far, far less than even one in one hundred. So if there is no compelling evidence for the hypothesis at all, its effect on the equation will be invisible. You can ignore it.

    In your more relevant discussion on pp. 204-205 (cited on p. 30 of On the Historicity of Jesus) you state

    The two hypotheses to test will be h=”Jesus was a historical person mythicized” and ~h= “Jesus was a mythical person historicized”. The only other logical possibilities are h0=”historical person not mythicized” and ~h=”mythical person not historicized”, but our background evidence firmly establishes the prior probability of either of those is vanishingly small…”

    Apart from the misuse of the negation operator, the first problem is that the hypothesis “Jesus was a historical person” can be asserted independently. Anything that follows “Jesus was a historical person [mythicized, made into legend, not mythicized, somewhat mythicized, mostly mythicized, not mythicized but equated with a mythicized character, etc.] is necessarily a further partition on the hypothesis space H1=”Jesus was a historical person”, and the only other logical possibility is ~H=”It is not the case that Jesus was a historical person”. The reason this matters is twofold:
    1) The hypotheses “Jesus was a historical person mythicized” and Jesus was a historical person not mythicized” are not two independent hypotheses. This is simple formal logic. To render it a bit more formally,let HP=”was a historical person”, “MP=”was a mythical person”, j=”Jesus”, M=”was mythicized”, and HZ=”was historicized”. We see that you have collapsed two predicates with each hypothesis, e.g. HPj & Mj. Letting x be anybody in particular, it is possible for “HPx & ~Mx” to be true, and therefore it is the predicates that are independent, not the hypotheses.
    Most importantly, the fact that predication is over the same variable/reference/etc. (i.e., Jesus) means that the truth of either hypothesis (“Jesus was mythicized” OR “Jesus was historical”) tells us something about the probability of the other. It is a priori more likely that, for any person x whatsoever, if HPx is true than ~HZx becomes less probable while Mx becomes more probable (assuming that one can “historicize” a historical person or “mythicize” a historical person). You not only conflate two logically distinct assertions about Jesus, you then partition the probability/hypothesis space H such that if we determine it is more probable Jesus was historical than that Jesus was mythical, it is still more likely that Jesus was mythical than that he was historical and not mythicized, despite the fact that one cannot be mythical and historical but can be historical and not mythicized. You render logically impossible that which is more probable.
    2) The partition on your hypothesis space consists of the entire set of mutually exclusive statements in H. For example:

    Definition 1 (Partition) A collection of sets {H1, . . . ,Hk} is a partition of another set H if
    1. the events are disjoint, which we write as Hi \ Hj = ; for i 6= j;
    2. the union of the sets is H, which we write as [formal notation I couldn’t type involving U with super- & subscripts]

    In the context of identifying which of several statements is true, if H is the set of all possible truths and {H1, . . . ,Hk} is a partition of H, then exactly one out

    of {H1, . . . ,Hk} contains the truth.

    Examples
    • Let H be someone’s religious orientation. Partitions include
    – {Protestant, Catholic, Jewish, other, none};
    – {Christian, non-Christian};
    – {atheist, monotheist, multitheist}.
    • Let H be someone’s number of children. Partitions include
    – {0, 1, 2, 3 or more};
    – {0, 1, 2, 3, 4, 5, 6, . . . }.
    • Let H be the relationship between smoking and hypertension in a given population. Partitions include
    – {some relationship, no relationship};
    – {negative correlation, zero correlation, positive correlation}.

    Partitions and probability
    Suppose {H1, . . . ,Hk} is a partition of H, Pr(H) = 1, and E is some specific event. The axioms of probability imply the following:
    [Rule of total & marginal probability as well as Bayes’ Rule]

    Examples
    A subset of the 1996 General Social Survey includes data on the education level and income for a sample of males over 30 years of age. Let {H1,H2,H3,H4} be the events

    that a randomly selected person in this sample is in, respectively, the lower 25th percentile, the second 25th percentile, the third 25th percentile and the upper 25th percentile in terms of

    income. By definition,
    {Pr(H1), Pr(H2),Pr(H3), Pr(H4)} = {.25, .25, .25, .25}.

    Note that {H1,H2,H3,H4} is a partition and so these probabilities sum to 1. Let E be the event that a randomly sampled person from the survey has a college

    education. From the survey data, we have
    {Pr(E|H1), Pr(E|H2),Pr(E|H3), Pr(E|H4)} = {.11, .19, .31, .53}.
    These probabilities do not sum to 1 – they represent the proportions of people with college degrees in the four different income subpopulations H1, H2, H3 and H4. Now

    let’s consider the income distribution of the college-educated population. Using Bayes’ rule we can obtain
    {Pr(H1|E), Pr(H2|E), Pr(H3|E), Pr(H4|E)} = {.09, .17, .27, .47} ,
    and we see that the income distribution for people in the college-educated population differs markedly from {.25, .25, .25, .25}, the distribution for the general population. Note that these

    probabilities do sum to 1 – they are the conditional probabilities of the events in the partition, given E.
    In Bayesian inference, {H1, . . . ,Hk} often refer to disjoint hypotheses or states of nature and E refers to the outcome of a survey, study or experiment. To compare

    hypotheses post-experimentally, we often calculate the following ratio:
    Pr(Hi|E) [divided by]
    Pr(Hj |E)
    =
    Pr(E|Hi)Pr(Hi)/ Pr(E) [divided by]
    Pr(E|Hj)Pr(Hj)/ Pr(E)
    =
    Pr(E|Hi)Pr(Hi) [divided by]
    Pr(E|Hj)Pr(Hj)
    =
    {Pr(E|Hi) [divided by]
    Pr(E|Hj)}
    ×
    {Pr(Hi) [divided by]
    Pr(Hj)}
    =
    “Bayes factor” × “prior beliefs” .
    This calculation reminds us that Bayes’ rule does not determine what our beliefs should be after seeing the data, it only tells us how they should change after seeing the data.

    Note that the reason for the total probability to be 1 is the partition of the probability space, but the probability measures are on the set of statements or events making up the partition. Thus once one has divided H into mutually exclusive and collectively exhaustive statements, events, hypotheses, etc., each one is given a probability measure (the “prior”) that is used in the final Bayesian model. This is a consequence of the way that Bayesian methods generalize BT by using it to update our uncertainty, represented by prior probability functions (or distributions). Because BT requires a partition of H to be collectively exhaustive and mutually exclusive, one and only one Hk is true, and our beliefs about each subset {H1, . . . ,Hk} change together. That’s why, in the example above, although the probabilities given E need not some to 1, the final collective posterior probabilities do.

    This is not always the case with Bayesian methods, but it is for the formula you use. To have your hypotheses be collectively exhaustive and mutually exclusive (as you require) means that the posterior probabilities must sum to one exactly as the prior probabilities do. One and only one hypothesis can be true, and it is not “rational” (in the Bayesian/decision-theoretic sense) to believe that out of a set of mutually exclusive and collectively exhaustive hypotheses the total posterior probability isn’t 1. If we believe that there are no other logical possibilities outside our partitioned set H, then whatever our information or evidence we use for our probability measures on each Hk in {H1, . . . ,Hk} which makes up the entirety of H, to believe that the posterior probabilities are together greater than 1 is nonsense (it is like believing the chances of getting heads or tails given a fair coin toss is e.g., 1.2) and to believe it is less is to admit that there is some other possibility and thus we didn’t, in fact, have a collectively exhaustive and mutually exclusive set.

    Testing two independent hypotheses requires a different mathematical approach. Going back to your example, lets assume (for the sake of simplicity of computation) something close to your “best odds on H“: a neat 30%. This is the probability that Jesus was BOTH historical AND mythicized given our evidence. Meanwhile, the probability, given our evidence, that Jesus was BOTH mythical AND historicized is 70%. Thus the probability that Jesus was mythicized can’t, in this model, be greater than 30%, while the probability that Jesus was historicized can’t be greater than 70% and likewise for Jesus being historical and mythical, respectively. However, your “partition” assumes the probability that Jesus was mythical and not historicized is vanishingly small. Let’s look at this a bit more formally, using that notations from above (HZ=”was historicized”, M=”was mythicized”, etc.). You’ve determined one possible value for the posterior probability P(MPj & HZ| evidence)= ~.7 or 70%. The probability that any two events occurred or any two hypotheses are true is AT MOST the less probable of the two or the minimum value. You’ve also determined, though, that P(MPj & ~HZj) = vanishingly small. If Jesus was likely mythical, then the reason for the vanishingly small probability P(MPj & ~HZj) must be because Jesus was almost certainly not historicized.

    • says

      As just shown above, you are way off the rails. You’ve screwed up the math from the first comment.

      I was saying, in effect:

      H1= J1 & J3 (“Jesus was historical” & “Jesus was mythicized”) => P(H1) = P(J1) x P(J3|J1) = 0.5 x 0.999999 = 0.4999995
      H2= J2 & J4 (”Jesus was mythical” & “Jesus was historicized”) => P(H2) = P(J2) x P(J4|J2) = 0.5 x 0.999999 = 0.4999995
      H3= J1 & ~J3 (“Jesus was historical” & “Jesus was not mythicized”) => P(H3) = P(J1) x P(~J3|J1) = 0.5 x 0.000001 = 0.0000005
      H4= J2 & ~J4 (“Jesus was mythical” & “Jesus was not historicized”) => P(H3) = P(J2) x P(~J4|J2) = 0.5 x 0.000001 = 0.0000005

      Total: 0.4999995 + 0.4999995 + 0.0000005 + 0.0000005 = 1

      And you have no argument against that. There is nothing inconsistent, incorrect, or invalid about this partition.

      So what are you going on about?

  21. messing says

    Sorry, my first attempt at trying to provide an intuitive notation failed doubly (it isn’t really intuitive and I kept getting it wrong). So, to correct it, let me reiterate that your hypotheses are conflations shown by basic formal logic/predicate calculus (and probability theory). Either Jesus was historical or Jesus wasn’t (this is a bit simplistic, but I’m trying as best as possible to work within the framework you set-up). A historical Jesus need not be mythicized in that for any individual, being historical doesn’t entail being mythicized (one can be historical and not be mythicized). The same is true for mythical and not historicized.

    Ideally, I’d like to represent any predicate using the standard single uppercase letter for the verb and a single lowercase for the subject, e.g., Mx= “x was mythicized”. However, unless I choose arbitrary letters I’ll end up with duplicates (e.g., Mx=”x was mythical” & “x was mythicized”). Instead, I’ll represent the hypothesis about Jesus as follows:
    J1=”Jesus was historical”
    J2=”Jesus was mythical”
    J3=”Jesus was mythicized”
    J4=”Jesus was historicized”
    H1= J1 & J3 (“Jesus was historical” & “Jesus was mythicized”)
    H2= J2 & J4 (”Jesus was mythical” &”Jesus was historicized”)
    H3= J1 & ~J3 (“Jesus was historical” & “Jesus was not mythicized”)
    H4= J2 & ~J4 (“Jesus was mythical” & “Jesus was not historicized”)

    Now, you determined from the beginning that P(H3) and P(H4) are vanishingly small. Recall that the probability of two hypotheses being true or two events/outcomes happening is AT MOST a probable as the lesser probability: if the probability that I’ll win the lottery tomorrow is one in a billion and the probability that it’ll rain tomorrow is .5, then the probability that it will BOTH rain AND I’ll win the lottery is AT MOST one in a billion. This means that, if J1 isn’t vanishingly small, then ~J3 must be. Likewise, if J2 isn’t vanishingly small, ~J4 must be. Now, presumably one can’t be historicized if one is historical and one can’t be mythicized if one is mythical. Therefore, if the probability that “Jesus was not mythicized” (~J2) is vanishingly small, the probability that “Jesus was mythical“ (J2) is vanishingly small. Likewise, if “Jesus was not historicized” (~J4) is vanishingly small, then the probability that “Jesus was historical” (J1) is vanishingly small. Hopefully, the paradox is alreadly apparent.
    I’ll use the same upper-end of the estimates you provided I did before (.3 & .7). If the probability of H2 is .7, this means the probability P(J2 & J4)=.7 and by the law for the conjunction of probabilities J2 & J4 must each have a probability of at least .7, because if either one had a lower probability than the combined probability would be lower than 70%. This means there is a non-vanishingly small probability that Jesus was historicized. Likewise, as the combined probability P(J1 & J3) is 30%, J1& J3 must BOTH have AT LEAST a .3 probability of being true. Thus all probabilities for J1, J2, J3, & J4 are not vanishingly small (in fact, they’re all AT LEAST 30%, and two are AT LEAST 70%). Which means it cannot be true that H3 & H4 are vanishingly small because if H3 were vanishingly small, either J1 must be (and isn’t) or ~J3 must be (which isn’t, because if the probability that “Jesus was not mythicized” were vanishingly small then the probability that Jesus was mythical would be too, as it would be almost certain that Jesus was mythicized and mythical individuals can’t be mythicized). Likewise, if H4 were vanishingly small, then either J2 must be as well (it isn’t) or ~J4 must be (and it isn’t, because if the probability that “Jesus was not historicized” were vanishingly small, Jesus must be historicized and historical people aren’t).

    So your partition and the laws of probability entail that your conclusions are paradoxical. Hopefully that’s clearer than the last mess I posted, although it would be nice if I could use graphics as the logic is, I’m sure, a bit hard to follow. It’s also kind of a moot point for reasons I’ve already mentioned, but it does illustrate the importance of clearly partitioning your probability/hypothesis space (not to mention being careful with formal or formal-ish notation, as I showed by confusing myself trying to be clear and failing to be clear or correct at the end of my last response.

    • says

      Okay. Now I’m getting to these comments.

      Hopefully, the paradox is alreadly apparent.

      Sorry, no.

      These are dependent, not independent, probabilities.

      It is only possible that ~J3 when J1. You can’t have ~J3 and ~J1. That’s logically contradictory.

      …if the probability that “Jesus was not mythicized” (~J3) is vanishingly small, the probability that “Jesus was mythical“ (J2) is vanishingly small.

      That makes no sense. “Not mythicized” is logically impossible if “mythical.” By definition if “mythical”, then “mythicized.”

      So you aren’t making any sense here.

      Here is how it actually works, with some dummy probabilities thrown in:

      J1=”Jesus was historical”
      J2=”Jesus was mythical”
      J3=”Jesus was mythicized”
      J4=”Jesus was historicized”

      H1= J1 & J3 (“Jesus was historical” & “Jesus was mythicized”) => P(H1) = P(J1) x P(J3|J1) = 0.5 x 0.8 = 0.4
      H2= J2 & J4 (”Jesus was mythical” & “Jesus was historicized”) => P(H2) = P(J2) x P(J4|J2) = 0.5 x 0.8 = 0.4
      H3= J1 & ~J3 (“Jesus was historical” & “Jesus was not mythicized”) => P(H3) = P(J1) x P(~J3|J1) = 0.5 x 0.2 = 0.1
      H4= J2 & ~J4 (“Jesus was mythical” & “Jesus was not historicized”) => P(H3) = P(J2) x P(~J4|J2) = 0.5 x 0.2 = 0.1

      Total: 0.4 + 0.4 + 0.1 + 0.1 = 1

      So much for your paradox.

      Can you really have not understood how this works?

      Now for the vanishingly small condition:

      H1= J1 & J3 (“Jesus was historical” & “Jesus was mythicized”) => P(H1) = P(J1) x P(J3|J1) = 0.5 x 0.999999 = 0.4999995
      H2= J2 & J4 (”Jesus was mythical” & “Jesus was historicized”) => P(H2) = P(J2) x P(J4|J2) = 0.5 x 0.999999 = 0.4999995
      H3= J1 & ~J3 (“Jesus was historical” & “Jesus was not mythicized”) => P(H3) = P(J1) x P(~J3|J1) = 0.5 x 0.000001 = 0.0000005
      H4= J2 & ~J4 (“Jesus was mythical” & “Jesus was not historicized”) => P(H3) = P(J2) x P(~J4|J2) = 0.5 x 0.000001 = 0.0000005

      Total: 0.4999995 + 0.4999995 + 0.0000005 + 0.0000005 = 1

      So, again, I ask, where is your paradox?

    • Anonymous Coward says

      I take it the central point is that Messing was treating J3, J4, ~J3, and ~J4 as independent which are actually dependent. The crux of your reply, if I understand correctly, comes when you map J3 on the left onto P(J3|J1) on the right (and so on for the remaining three rows mapping hypotheses to probabilities). That seems to be what you’re saying Messing missed. Is that right?

    • says

      It’s hard to say what messing was thinking because nothing he says makes much sense. (There was also a typo in my last, which I just fixed; the last expression is P(J2) x P(~J4|J2), not P(J2) x P(~J4|J1).)

      It sounds like, maybe (?), he thought it was possible that ~Jesus, and therefore we need to account for that. But ~Jesus entails ~Christianity, and Christianity is in b (its existence is well-established background knowledge), which means P(~Jesus|b) = P(~Jesus|Christianity), but P(~Christianity|Christianity) = 0, and if ~Jesus, then ~Christianity, therefore P(~Jesus|Christianity) = 0. Therefore, using his mistaken reading, P(~J1.~J3|b) = 0.

      Or the mistake he is making might instead be that he doesn’t realize J2 is ~J1.

      Or that he doesn’t realize that “mythicizing” can only occur to a historical person, and “historicizing” can only happen to a mythical person (which is also related to the dependent probability issue).

      Another way of stating it is:

      J1=”Jesus was historical”
      ~J1=J2=”Jesus was mythical”
      J3=”If Jesus was historical, he was mythicized”
      and ~J3=”If Jesus was historical, he was not mythicized”
      J4=”If Jesus was mythical, he was historicized”
      and ~J4=”If Jesus was mythical, he was not historicized”

      It then becomes clear that you can never have J1 & J4, only ~J1 & J4 and ~J1 & ~J4. Likewise you can never have ~J1 & J3, only J1 & J3 and J1 & ~J3.

      That’s why (for example):

      H1= J1 & J3 (“Jesus was historical” & “Jesus was mythicized”) => P(H1) = P(J1) x P(J3|J1) = 0.5 x 0.8 = 0.4
      H2= ~J1 & J4 (”Jesus was mythical” & “Jesus was historicized”) => P(H2) = P(~J1) x P(J4|~J1) = 0.5 x 0.8 = 0.4
      H3= J1 & ~J3 (“Jesus was historical” & “Jesus was not mythicized”) => P(H3) = P(J1) x P(~J3|J1) = 0.5 x 0.2 = 0.1
      H4= ~J1 & ~J4 (“Jesus was mythical” & “Jesus was not historicized”) => P(H3) = P(~J1) x P(~J4|~J1) = 0.5 x 0.2 = 0.1

      So we bifurcate the area between J1 and ~J1, then within each division we demarcate by either J3 (J1) or J4 (~J1), getting J1.J3 and J1.~J3 on the one side, and ~J1.J4 and ~J1.~J4 on the other side. We don’t demarcate both sides by J3 because P(~J1|J3) = 0. And we don’t demarcate both sides by J4 because P(J1|J4) = 0.

      In even simpler terms, J3 is a predicate of J1, not J2, and J4 is a predicate of J2, not J1. Thus you can have “If J1, then either J3 or ~J3,” which demarcates the entire space of possibilities for J1, and “If J2, then either J4 or ~J4,” which demarcates the entire space of possibilities for J2. And J1 and J2 demarcate the entire space of possibilities.

      And that’s it.

      How messing fails to grasp that, I don’t know.

  22. Anonymous Coward says

    I see here now what you were talking about concerning h and -h.

    First, I think there’s a crucial typo in what you wrote which left me confused for a long while. You said,

    “Therefore, if the probability that “Jesus was not mythicized” (~J2) is vanishingly small, the probability that “Jesus was mythical“ (J2) is vanishingly small. Likewise, if “Jesus was not historicized” (~J4) is vanishingly small, then the probability that “Jesus was historical” (J1) is vanishingly small.”

    I think two of those “vanishingly small”s were supposed to have “not”s prepended to them!

    Assuming I’m right about that, you’re arguing that though Carrier said H3 and H4 are vanishingly small, they actually can’t be vanishingly small since J1, J2, J3, and J4 all fail to be vanishingly small. And for J3 and J4, your argument that they’re not vanishingly small is that ~J3 and ~J4 must (by hypothesis) be vanishingly small given that J1 and J2 are not, and given that J1 & ~J3 is (by hypothesis) vanishingly small, and also J2 & ~J4 is (by hypothesis) vanishingly small.

    I think the thrust of your point can be put succinctly as this. If H3 and H4 are vanishingly small, then it follows that J3 and J4 are both almost certain! In other words it follows that it is almost certain that Jesus was historicized AND that he was mythicized.

    I don’t know how to answer this, I’ll have to wait to see how Carrier can reply.

  23. Anonymous Coward says

    I think I may have misconstrued your argument in my previous post–I can’t read it now as it’s awaiting moderation, so I can’t check. On rereading your argument, it occurs to me that it’s not clear that you can’t historicize a historical individual or mythicize a mythical individual. It depends on what is meant by “historicize” and “mythicize.” If “historicize” means “place in a historical setting without regard for accuracy,” you can historicize a historical individual. And if mythicize means “make mythic claims about the individual without following limits suggested by stories already told about the individual,” then you can mythicize a mythical individual.

    If what Carrier means by historicize and mythicize allows for historicizing historical individuals and mythicizing mythical individuals, then your criticism may not hit the mark, as it relies on the premise that you can’t mythicize a mythical individual and can’t historicize a historical individual.

    Anyway, something else occurs to me. Couldn’t the whole argument simply be run on J1 and J2, forgetting about J3 and J4 entirely? J1 and J2 themselves completely partition the space between the two of them.

    • says

      [Just FYI, there were no other comments by messing awaiting moderation when you posted the above. Everything he submitted had been posted. At least so far as got to my queue.]

    • messing says

      This is absolutely true. If one can “historicize” an historical individual, then the entirety of my attempt to use Dr. Carrier’s framework to demonstrate what is meant by probability space/hypothesis space/sample space and partition is wrong. Of course, it also further makes Dr. Carrier’s partition ridiculous as one who is “historical” can clearly be “historicized”, yet this isn’t an proffered option. It isn’t even one of the “vanishingly small” probabilities.

      However, if we ignore this and consider the probability that “Jesus was an historical individual historicized” and understand historicize to mean “place in a historical setting without regard for accuracy” than this differs little from “make mythic claims”. From The Iliad to modern historical fiction, “historicize” becomes a matter of subjective determination that some setting is “historical”. For example, regardless of whether Jesus was historical, the four “canonical” gospels as well as numerous other sources situate him within a real, historical region (Galilee & Jerusalem), reference “historical” figures such as Pilate, refer to real time periods, etc. It isn’t set in Middle Earth or some other fantastical realm. The same is true of Hesiod and Homer’s works (or rather, those attributed to these figures). Troy, Ithaca, the Achaeans, etc., are historical referents. Dr. Carrier, in his comparison of Socrates and Jesus blatantly overlooks the modern realization that our sources for Socrates have been critically examined and doubted since at least Garnier’s 1768 critique of Xenophon (and in which he cites the Blackwell companion to Socrates and ignores the fact that from Dupréel & Gigon to a contribution within that volume by Dorion, the “Socratic problem” has faced continual problems and resulted in the very kind of extreme skepticism that he himself has of Jesus’ existence), and that both Aristotle and Diogenes Laertius categorize our central sources for Socrates as a kind of fiction (Diogenes Laertius, in fact, tells us some ~6 centuries after Socrates that the man himself happened upon someone reading Plato’s work and exclaimed “what lies that lad tells about me!”). So if “historicize” is used to have meaning, it seems natural that it mean “to make historical that which is of the mythical/legendary/fictional”.

      Of course, it doesn’t really matter, as if one can “historicize” historical figures than this is yet another failure on Dr. Carrier’s part to partition the probability space in a way required for the application of Bayes’ theorem.

    • says

      This is all silly.

      I do not use “historicize” in this fashion.

      Historicizing (a person) means making historical (a person) who previously was not.

      Period.

      Any other definition is simply ignoring what I say and do in Proving History and OHJ.

      So if that’s what your criticism is about, you’ve gone of the rails from the start.

  24. lpetrich says

    You state “Because Superman also fits the Rank-Raglan profile, as do Anakin Skywalker, Optimus Prime, Aragorn, and Captain Kirk: all score above 10″, quoting from that flairicus.com blog entry. I went there, and I couldn’t find how these scores were calculated. I’m especially suspicious of the Captain Kirk one, since in the Star Trek canon, his origins and early life were very uneventful. I did a lot of further searching, with a little bit more success.

    The Science Pundit: Raglan’s Scale and the Hero of Star Wars He scores Anakin Skywalker at 18 and Luke at 6, though Luke’s biography is incomplete.

    Tsar Nicholas II John Nagle found that he scores very high for someone well-documented: 14.

    I myself have scored several people, though my scores are sometimes rather different from the scores that Lord Raglan and others had found. I find Moses 15, King David 4, Jesus Christ 19.5, Krishna 17, The Buddha 13, Zeus 14.5 of 16, Hercules 15, Perseus 17, Oedipus 13, Romulus 19, Alexander the Great 9, Julius Caesar 9.5, Augustus Caesar 10, King Arthur 14.5, George Washington 6, Napoleon 8, Abraham Lincoln 6, Charles Darwin 5, Winston Churchill 5, Adolf Hitler 4, JFK 8 (with conspiracy theories), Muammar Gaddafi 5.5, Anakin Skywalker 10.5, Luke Skywalker and Leia Organa 10 of 15, Harry Potter 12 of 15.

    After doing lots of scores, one recognizes some patterns. The origins and early years of most well-documented heroes are usually very uneventful, and they often come to power in relatively undramatic ways. They often have rather dramatic careers while in power, and they are seldom repudiated. I scored Muammar Gaddafi because he was rather clearly repudiated. The only other repudiated ones I could think of were Richard Nixon and Mikhail Gorbachev, though Tsar Nicholas II also qualifies. Napoleon and Hitler I concluded were overwhelmed rather than repudiated. Well-documented heroes often die of old age in their homes, a very undramatic way to go. The exceptions include some subterranean ones: Nicholas II in the house’s basement, Hitler in his bunker, Gaddafi discovered in a storm drain. Was there ever a legendary hero who spent the last moments of his life cowering in a cave?

    • says

      All of which is addressed in chapter six of the book.

      If RR scores were arbitrary, there should be as many historical persons scoring high as score low, when the number scoring high is significant (e.g. more than ten members). That there aren’t is an objective fact of the world that must be taken into account. Period. You can’t dodge reality.

      As to Hallquist’s attempt to downscore Jesus, Jesus scores 14 on Mark alone. Add the nativities and he almost maxes out. Hallquist is mistreating the data to try and force the result he wants. Indeed, he acts like a Christian fundamentalist:

      “7. He is reared in a foreign country by one or more foster parents” — Joseph is a foster parent by definition, i.e. he is not the physical father of Jesus, and that’s “one or more,” and Jesus is reared in a foreign country: he remains exiled from Judea for over ten years. Indeed, forcing Jesus to leave his home town and even be taken to Egypt by a father who isn’t really his is blatant mythmaking; for Hallquist to think this should not be counted as a deliberate attempt to fit him to the profile is rather desperate.

      “9. On reaching manhood he returns to his future kingdom” — Hallquist bizarrely treats this as literally as a fundamentalist (as if the profile required returning to his kingdom literally at the very moment of reaching manhood; no, it means after becoming an adult, as in, he does not return to his kingdom as a child).

      And Hallquist, again like a fundamentalist, handwaves about Jesus not being identical to other king tales, as if that was relevant to the criteria:

      “10. He is crowned, hailed or becomes king.” — Yes. The whole triumphal entry is explicitly hailing him king. And is likewise explicit mythmaking.
      “11. He reigns uneventfully (i.e., without wars or national catastrophes).” — Exactly what happens.
      “12. He prescribes laws.” — Exactly what happens.
      “13. He then loses favor with the gods or his subjects.” — Exactly what happens (indeed infamously and conspicuously so).
      “14. He is driven from the throne or city.” — He is literally driven out of the city (Mark 15:20-22) to be killed.
      “15. He meets with a mysterious death.” — Here I can only assume Hallquist didn’t check the other examples, and thus mistakenly thinks this only means “death by unknown means,” when in fact it means “any death of a mysterious character,” and here upon his death, which is mysteriously sudden (Mk. 15:44), mysteriously causes the sun to mysteriously go out for three hours and the (80ft. high) temple curtain to mysteriously tear asunder. That’s a mysterious death.

      By contrast, I shall note again that Hallquist still does not show Kim Jong-Il even scores high enough on the Rank-Raglan scale to count. He is unreasonably strict when scoring Jesus, but unreasonably liberal when scoring Kim. Just like a Christian fundamentalist.

      But what is most bizarre is that now Hallquist complains that I am too generous in counting historical persons in the class, while simultaneously complaining that I am undercounting historical persons in the class. WTF?

      So now not only does he not know how a probability works (despite his protestations, he did indeed hose this the first time around), he also can’t even get his complaints straight.

      This is one of the most blatant examples of motivated reasoning I’ve ever seen. He can’t even be logical, so desperate is he to deny the facts.

      Indeed, he still illogically claims “If Carrier were right that no other real historical figures have ever fit 12 or 13 out of the 22 criteria, that would definitely be a point in his favor.” And that after I just explained to him (!) that even if there were four real historical figures scoring as high, we’d still end up with my same prior probability. Again, he does not appear to actually know how a probability works. It’s either all one or all the other for him. That it’s a frequency seems beyond his ability to grant. Why?

      Indeed, he repeats this black and white thinking and even falsely projects it onto me: “Carrier seems to want everything to be either totally irrelevant, or else a decisive point in his favor.” Uh, no. I’m the one who is saying it’s not a decisive point, that there could be as many as four out of the fourteen high scoring heroes who were historical. I’m being extremely generous to the alternative. Hallquist is the one who refuses not to be.

      This is all very perplexing. And not a little embarrassing.

    • Bernard says

      Hi Dr. Carrier,
      .
      Here is my comment where I disagree with you on this Rank-Raglan scale for Jesus according to Matthew’s gospel:
      .
      9 On reaching manhood he returns to his future kingdom.
      .
      Actually, well after he became an adult, he went to Jerusalem (not really returned) and as long as he was still alive, that was never his future kingdom. So a big 0 on that one.
      .
      11 He reigns uneventfully (i.e., without wars or national catastrophes).
      .
      But he did not reign whatsoever in Jerusalem. He never was a ruling king in Jerusalem during his lifetime. He was hailed as a king outside Jerusalem but that’s about it.. Another 0
      .
      14 He is driven from the throne or city.
      .
      How could he be driven from a throne if he did not have any? And “Matthew” did not even say that the crucifixion was outside the city. It might be implied in Mark’s gospel (15:21) but not in Matthew’s gospel. More, the opposite is stated: Jesus is driven from outside the city to within it after his arrest. So another 0 for that one. I would even go for -1 if it was allowed.
      .
      16 He dies atop a hill or high place.
      .
      Where does it say Jesus was crucified on a hill in Matthew’s gospel? And Golgotha means place of the skull, not hill of the skull. Another 0 for that one.
      .
      17 His children, if any, do not succeed him.
      .
      He had no children, and no throne to be succeeded from, so the result had to be a 0
      .
      19 Yet he still has one or more holy sepulchers (in fact or fiction).

      If his body disappeared, how could he have a sepulcher? The fact that a tomb was selected as a place of worship way later, does not count because not in Matthew’s gospel. Another 0
      .
      20 Before taking a throne or a wife, he battles and defeats a great adversary
      (such as a king, giant, dragon or wild beast).
      .
      Well, again Jesus never took a throne or a wife. And the temptations from the devil/Satan cannot be described as a battle, and the devil/Satan is not defeated but just leave him. Another 0
      .
      On the two last points, I agree with you for a 0 of both counts.

      So the score for Jesus according to the gospel of Matthew is 22 – 9 = 13 not 20
      .
      And for Mark’s gospel, that would be 8 not 14.
      Changes from my rating of Matthew’s gospel:
      0 for 1, 2, 3, 5, 6, 7 1 for 14
      .
      I think you have grossly exaggerated your rating.
      .
      Cordially Bernard

    • says

      Bernard, you are a crank.

      That has been well-established, in case anyone new here doesn’t know this.

      Returns to future kingdom: The entire Triumphal Entry story is a paradigmatic example of a king returning to his kingdom. This is admitted by all mainstream scholars. And that’s even in Mark, not just Matthew. The criterion has nothing whatever to do with this happening “immediately” upon hitting puberty. You are acting like a Christian apologist and ignoring the data and making a fuss over hyper-specific definitions no one but you uses…which we call “Straw Man” argumentation.

      He reigns uneventfully: He was hailed a king by the people, and even declared so by the Roman governor (As Jesus points out: “You said it”; it’s even officially written above him on the cross). Only Christian fundamentalists (and cranks like you) read texts absurdly hyper-literally. The rest of us can plainly see Jesus was hailed king by the people, and the remaining authorities considered illegitimate. When Jesus proclaims laws, he is doing so in his capacity as the true king of Israel, telling everyone about his kingdom. This is undeniably the entire point even in Mark, much less Matthew.

      He is driven from the throne or city: He loses favor from the people who hailed him king, and is thus condemned to be crucified by them. In result he loses his reign, is driven outside the city by the mob, and killed. That’s a hit. Only a fundie or crank would not think so. This is how literary allusion works. It’s not ridiculously absurdly hyper-literal. But it’s still obvious to everyone except to fundies and cranks.

      And “Matthew” did not even say that the crucifixion was outside the city.

      Yes, he does. Mt. 27:31-33. It was common knowledge that crucifixions were held “outside” the city, not just outside the court, and Matthew’s intended Jewish readers certainly knew that Golgotha was a hill outside the city.

      Only cranks try to turn that into a claim that Matthew relocated the crucifixion inside Jerusalem.

      Where does it say Jesus was crucified on a hill in Matthew’s gospel?

      Golgotha was well known to be a hill. Every Christian author of the era knew this.

      Just because you are a 21st century foreigner doesn’t allow you to project your ignorance onto a people who knew their world a far sight better than you do.

      But more to the point, the word (as all Gospels say) refers only to the top part of the skull (the kranion, not the skyphion)…in other words, the name is a description of the place: a hill.

      He had no children, and no throne to be succeeded from, so the result had to be a 0.

      Since he could have had children, who could indeed have succeeded him (hailed by the people as their true king, or headed the church, explicitly referred to as his kingdom), your statement is false.

      If his body disappeared, how could he have a sepulcher?

      Weird, huh? And yet, he does…even in the Gospels (all of them, even, gasp!, Matthew).

      Well, again Jesus never took…a wife.

      He could have. Therefore, that he didn’t, scores.

      And the temptations from the devil/Satan cannot be described as a battle, and the devil/Satan is not defeated but just leave him.

      Like a fundie, being a crank, you again can’t see anything except hyper-literally. Sorry, literature isn’t absurdly ridiculously hyper-literal.

      Oedipus defeats the Sphinx by answering a riddle.

      This is another case where your embarrassingly irresponsible refusal to read and respond to the book you claim to be criticizing makes you look like an utter fool to everyone who has read it (p. 233).

      The Gospels clearly regarded the Temptation as a spiritual battle. Jesus confronts and defeats the Devil. He overcomes his temptations and is triumphant. Consequently the Devil has no power over him.

      22 – 9 = 13 not 20

      Even were that true, that’s still a ranking score. It’s above half. So it doesn’t even get Jesus out of the reference class!

      (And as I point out, even in Mark Jesus scores 14…that’s actually, not according to crank-brained literary reading methods.)

      Bernard, just go away.

      You are wasting everyone’s time with your pseudohistorical nonsense and irresponsibility.

    • Bernard says

      Dr. Carrier,
      .
      RC: “Returns to future kingdom: The entire Triumphal Entry story is a paradigmatic example of a king returning to his kingdom. This is admitted by all mainstream scholars. And that’s even in Mark, not just Matthew. The criterion has nothing whatever to do with this happening “immediately” upon hitting puberty. You are acting like a Christian apologist and ignoring the data and making a fuss over hyper-specific definitions no one but you uses…which we call “Straw Man” argumentation.”
      .
      BM: You are interpreting, as apparently all (unnamed) mainstream scholars (why don’t you name a few since you obviously know what all mainstream scholars wrote on the matter). But how do you know “Matthew” wanted the triumphal entry interpreted that way?
      If he did, he would have indicated Jesus had a kingdom based on Jerusalem (to return to).
      If he did, he would have indicated that Jesus went to Jerusalem prior to his last days, so a return of some sort.
      But none of the above.
      And “Matthew” had Jesus going to Jerusalem well after he reached manhood.
      .
      RC: “He reigns uneventfully: He was hailed a king by the people, and even declared so by the Roman governor (As Jesus points out: “You said it”; it’s even officially written above him on the cross). Only Christian fundamentalists (and cranks like you) read texts absurdly hyper-literally. The rest of us can plainly see Jesus was hailed king by the people, and the remaining authorities considered illegitimate. When Jesus proclaims laws, he is doing so in his capacity as the true king of Israel, telling everyone about his kingdom. This is undeniably the entire point even in Mark, much less Matthew.”
      .
      BM: Being hailed as somebody does not mean the one being hailed thinks that or pretends to be that.
      For example, you hail me as a crank (or if you would hail me as a genius), but that ‘s not what I think of myself or pretend to be.
      “Matthew” had Jesus, just like “Mark”, saying “you said it”, but at 27:23, Pilate does not consider that answer as an admission Jesus is king of the Jews, because he thinks Jesus committed no crime. And one thing is certain: “Matthew” never wrote Jesus reigned during his days in Jerusalem.
      As far as proclaiming laws. “Matthew” had not Jesus doing that. The law that Jesus wants for Jewish Christians to follow is the whole law of Moses (8:17-18).
      But most orthodox Jews would want the same for other Jews. In other words, you did not have to be a king of the Jews to say the entire law should be kept.
      .
      RC: “He is driven from the throne or city: He loses favor from the people who hailed him king, and is thus condemned to be crucified by them. In result he loses his reign, is driven outside the city by the mob, and killed. That’s a hit. Only a fundie or crank would not think so. This is how literary allusion works. It’s not ridiculously absurdly hyper-literal. But it’s still obvious to everyone except to fundies and cranks.”
      .
      BM: you are combining two Rank-Raglan points together:
      I disagreed with you on point 14 “He is driven from the throne or city.” but not on point 13 “He then loses favor with the gods or his subjects.”
      Again there was no reign to loose, no throne to be led away, and “Matthew” never even suggested (unlike “Mark”) that the crucifixion was outside the city.
      .
      RC: ““And “Matthew” did not even say that the crucifixion was outside the city.”
      Yes, he does. Mt. 27:31-33. It was common knowledge that crucifixions were held “outside” the city, not just outside the court, and Matthew’s intended Jewish readers certainly knew that Golgotha was a hill outside the city.
      Only cranks try to turn that into a claim that Matthew relocated the crucifixion inside Jerusalem.”
      .
      BM: Mt 27:31-33:
      “And when they had mocked him, they stripped him of the robe, and put his own clothes on him, and led him away to crucify him.
      As they went out, they came upon a man of Cyre’ne, Simon by name; this man they compelled to carry his cross.
      And when they came to a place called Gol’gotha (which means the place of a skull),”
      “out”, according to the context, is outside the praetorium (27:27), not outside the city.
      .
      RC: ““Where does it say Jesus was crucified on a hill in Matthew’s gospel?”
      Golgotha was well known to be a hill. Every Christian author of the era knew this.
      .
      Just because you are a 21st century foreigner doesn’t allow you to project your ignorance onto a people who knew their world a far sight better than you do.
      .
      But more to the point, the word (as all Gospels say) refers only to the top part of the skull (the kranion, not the skyphion)…in other words, the name is a description of the place: a hill.”
      .
      BM: How do you know Golgotha was a hill, and not a small rocky outcrop? and every Christians of the era knew? Evidence please.
      If “Matthew” wanted to add up another so-called mythological element, he would have specified a hill. He did not (as every other gospel authors, even “John” who seems to know about pre-70 Jerusalem).
      .
      RC: ““He had no children, and no throne to be succeeded from, so the result had to be a 0.”
      Since he could have had children, who could indeed have succeeded him (hailed by the people as their true king, or headed the church, explicitly referred to as his kingdom), your statement is false.
      .
      BM: The Rank-Raglan point is “His children, if any, do not succeed him.” He had no children, so this point do not apply, so zero. Why do you hypothesize he could have had children, and then hypothesize again from that hypothesis these children could indeed have succeeded him? And then, score a 1?
      .
      You want to rate Jesus as described by “Matthew”, but many times you go outside his gospel to find evidence, assumption or hypothesis you need when there is no evidence in the gospel in order to support your case.
      .
      RC: ““If his body disappeared, how could he have a sepulcher?”
      .
      Weird, huh? And yet, he does…even in the Gospels (all of them, even, gasp!, Matthew).”
      .
      BM: Early Christians did not have a holy sepulcher to go to (contrary to Osiris, who had many). The holy sepulcher “discovery” and worship came much later.

      RC: ““Well, again Jesus never took…a wife.”
      .
      He could have. Therefore, that he didn’t, scores.”
      .
      BM: Confusing statement: but it looks to me. “he could have had a wife” (a supposition) is equivalent to, in your view, “he had a wife”. If it is so, that’s unacceptable by me.
      .
      RC: ““And the temptations from the devil/Satan cannot be described as a battle, and the devil/Satan is not defeated but just leave him.”
      Like a fundie, being a crank, you again can’t see anything except hyper-literally. Sorry, literature isn’t absurdly ridiculously hyper-literal.
      Oedipus defeats the Sphinx by answering a riddle.
      .
      This is another case where your embarrassingly irresponsible refusal to read and respond to the book you claim to be criticizing makes you look like an utter fool to everyone who has read it (p. 233).
      .
      The Gospels clearly regarded the Temptation as a spiritual battle. Jesus confronts and defeats the Devil. He overcomes his temptations and is triumphant. Consequently the Devil has no power over him.”
      .
      BM: Oedipus really defeats the Sphinx, because after answering correctly the riddle, the Sphinx, in rage, kills himself. This is not the case of the devil/Satan after the temptation. Satan is still alive and well, just that he failed to corrupt Jesus. Later, in the gospel, Satan is victorious by having (through Judas) Jesus arrested, which led to the crucifixion.
      .
      Cordially, Bernard

    • says

      This isn’t even worth a response, Bernard. This is just endless crank bullshit. Which is what you always do. Over and over and over again. You just answer every refutation by dumping tens of thousands of words repeating the same crank, quasi-fundamentalist garbage. This is a waste of everyone’s fucking time.

      Go. Away.

  25. Psychopomp Gecko says

    While I agree with you overall that there was most likely either no historical Jesus or a guy who was almost completely unlike the gospels, I can’t say I’ll be buying the book. It appears to have a lot of this Bayesian stuff in it and all the probability puts it outside where I’m going to read. It’s enough for me to know that there are no credible contemporary sources to justify believing the claim “Jesus existed”.

    The contemporary comes from the fact that every source outside the bible was written decades after the fact, and the credible comes from the Biblical sources which also believed that donkeys were capable of speech.

    That said, I hope you keep up the good work so I can finally change those deceptively-edited Wikipedia pages about the Historicity of Jesus.

    • says

      It’s enough for me to know that there are no credible contemporary sources to justify believing the claim “Jesus existed”.

      To be fair, note that this is disputed. Sometimes foolishly (e.g. Josephus), yet nevertheless my book covers why that’s foolish (without involving any explicit math, and extensively citing scholarship). Sometimes more soberly (e.g. the Epistles of Paul), where again my book engages the reasons for ultimately rejecting the conclusion of historicity (without involving any explicit math, and extensively citing scholarship). But in the end you have to put numbers to things. If you want to say something is improbable or probable, you have to identify what you mean by that. Otherwise you mean nothing by it.

      So there is no avoiding the math. The best we can do is conceal it behind dodgy words. Which we shouldn’t desire to do, IMO.

    • Anonymous Coward says

      Richard may not like this approach, (I’m not sure,) but to be honest you can skim the math. The book is extremely valuable simply as a repository of information you might not find collected together anywhere else in a work easily accessible to laymen.

    • Geoff says

      The meat is really in the discussion of priors, the math is just the cherry on top. Richard did what he said he would do; he showed how BT requires the historian to examine carefully underlying assumptions and expose the foundational logic in a transparent fashion. Commentators like F. Ramos who complain about the lack of math in the book are simply misguided. The entire book is an examination and explanation for the numbers that Carrier plugs into the equation. It is 700 pages of avoiding the garbage in-garbage out complaint. What Richard put in was not garbage and he demonstrates that throughout. I have not seen any reviews or critiques that have successfully altered Richard’s priors, even on the most vulnerable points, such as Galatians 1:19. I have said before, this book is the standard now for the mythicist position and, I would argue, the ball is in the historicist camp right now.

  26. messing says

    Here is how it actually works, with some dummy probabilities thrown in:

    J1=”Jesus was historical”
    J2=”Jesus was mythical”
    J3=”Jesus was mythicized”
    J4=”Jesus was historicized”

    Do you know what “independent” vs. “dependent” means in terms of probability? You are so bereft of mathematical knowledge and familiarity with basic probability theory that “probability space” is some exotic phrase for you, so I can only wonder whether you are simply using some common parlance interpretation here.

    Historicizing (a person) means making historical (a person) who previously was not.

    That’s what I thought. Which makes your “partition” utterly nonsensical and your “logic” paradoxical. You fail to understand this because you unify distinct propositions in contradiction with all of formal logics, mathematics, and probability:

    H2= ~J1 & J4 (”Jesus was mythical” & “Jesus was historicized”) => P(H2) = P(~J1) x P(J4|~J1) = 0.5 x 0.8 = 0.4

    First, once again you abuse basic notation by failing to properly use the negation operator. Second, if “historicizing” a person means making historical a person who previously wasn’t, then to historicize Jesus is to make historical that which by definition was not. If Jesus was historicized, he wasn’t historical. If your grasp of probability and logic is so weak that you fail to realize the connection between making an individual “historical” and realizing that this entails they weren’t, then you shouldn’t be pretending to be a philosopher, let alone capable of logical and/or Bayesian analysis. If Jesus were historicized, then clearly he was (under the most generous reasoning) almost certainly mythical and not historical. However, your inability to appropriately use notation or understand even that you use somehow gives you the idea that the probability that Jesus was historicized AND mythical is less than .5. This is idiocy. If Jesus were historicized, it means (according to you) making historical an individual who wasn’t. Unless your understanding of mythical is utterly idiomatic, this means making historical what was mythical (or completely unknown). You just plug numbers into nonsense and pretend they mean something.

    So, again, I ask, where is your paradox?

    It’s called entailment and probability theory. Look it up. When you know enough to correctly use the most basic operators in formal logic, then maybe we can get somewhere. I separated logical propositions you combined. By combining them, your logic leads to paradoxes because of the relationships between the real propositions underlying your “analysis”. This is related to partitioning hypothesis/probability space. You still don’t know what this means. This betrays a basic, fundamental lack of that which you seek to explicate.

    • says

      I am baffled. You are not even talking about what I actually demarcated. I am having a hard time even finding anything intelligible in what you are saying. I am starting to feel concerned about your mental health.

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